Undefined Te Terms and Intuitive Concepts of Geometry Undefined terms: In geometry, definitions are formed using known words or terms to describe a new word. There are three words in geometry geometry that are not formally defined. These three undefined undefined terms terms are point, line and plane.
POINT (an undefined term) In geometry, a point has no dimension (actual size). Even though we represent a point with a dot, the point has no length, width, or thickness. ur dot can be very v ery tiny or very large and it still represents a point. ! point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, x,y). (x,y).
LINE (an undefined term) In geometry, a line has no thickness but its length e"tends in one dimension and goes on forever in both directions. #nless otherwise otherwise stated a line is drawn as a straight line with two arrowheads indicating that the line e"tends without end in both directions. directions. ! line is named by a single lowercase letter, points on the line, .
, or by any two
PLANE (an undefined term) In geometry, a plane has no thickness but e"tends indefinitely in all directions. $lanes are usually represented by a shape that looks like a tabletop or a parallelogram. Even though the the diagram of a plane has edges, you must remember that the the plane has no boundaries. ! plane is named by a single letter (plane m) or by three non%collinear points (plane !&').
Intuitive Concepts: There are a few basic concepts in geometry that need to be understood, but are seldom used as reasons reasons in a formal proof.
Collinear Points
points that lie on on the same line.
Coplanar points
points that lie in in the same plane.
Opposite rays
rays that lie on the same line, with a common endpoint and no other points points in common. common. pposite rays form a straight line andor a straight angle (*+-).
Parallel lines
two coplanar lines that do not intersect
Se! lines
two non%coplanar lines that do not intersect.
Consider t"e t"e follo!in# t"eorems t"eorems relatin# relatin# lines and planes$ A dia#ram is supplied for eac" t"eorem t"at repre represents sents one possi%le depiction of t"e situation$ If a lin linee is is per perpe pend ndic icul ular ar to each each of two two
Thro Throug ugh h a give given n poi point nt ther theree pas passe sess one one and and
intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
only one plane perpendicular to a given line.
Through a given point there passes one and only one line perpendicular to a given plane.
Two Two lines perpendicular to the same plane are coplanar.
Two planes planes are are perpe perpendi ndicul cular ar to each each othe otherr if
If a line line is perpen perpendic dicula ularr to a plane plane,, then then any
and only if one plane contains a line perpendicular to the second plane.
line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
If a plane intersects two parallel planes, then the intersection is two parallel lines.
If two planes are perpendicular to the same line, they are parallel.
The angle where two planes meet is called a dihedral angle. Woodworkers and construction workers deal with dihedral angles. For example, creating a rafter for a hip roof requires an understanding of dihedral angles.
Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (// &.'.) Euclid0s te"t Elements was the first systematic discussion of geometry. 1hile many of Euclid0s findings had been previously stated by earlier 2reek mathematicians, Euclid is credited with developing the first comprehensive deductive system. Euclid0s approach to geometry consisted of proving all theorems from a finite number of
postulates (a"ioms). Euclidean 2eometry is the study of flat space. 1e can easily illustrate these geometrical concepts by drawing on a flat piece of paper or chalkboard. In flat space, we know such concepts as
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the shortest distance between two points is one uni3ue straight line.
the sum of the angles in any triangle e3uals *+ degrees.
the concept of perpendicular to a line can be illustrated as seen in the picture at the right.
In his te"t, Euclid stated his fifth postulate, the famous parallel postulate, in the following manner If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if e"tended indefinitely, meet on that side on which are the angles less than the two right angles. Today, we know the parallel postulate as simply stating Through a point not on a line, there is no more than one line parallel to the line.
The concepts in Euclid0s geometry remained unchallenged until the early *4th century. !t that time, other forms of geometry started to emerge, called non% Euclidean geometries. It was no longer assumed that Euclid0s geometry could be used to describe all physical space.
non&Euclidean #eometries:
are any forms of geometry that contain a postulate (a"iom) which is e3uivalent to the negation of the Euclidean parallel postulate.
Examples: '$ (iemannian Geometry
(also called elliptic #eometry or sp"erical #eometry) ! non%Euclidean geometry using as its parallel postulate any statement e3uivalent to the following If l is any line and P is any point not on l , then there are no lines through P that are parallel to l .
5iemannian 2eometry is named for the 2erman mathematician, &ernhard 5iemann, who in *++4 rediscovered the work of 2irolamo 6accheri (Italian) showing certain flaws in Euclidean 2eometry.
5iemannian 2eometry is the study of curved surfaces. 'onsider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface, such as a sphere. The study of 5iemannian 2eometry has a direct connection to our daily e"istence since we live on a curved surface called planet Earth.
1hat effect does working on a sphere, or a curved space, have on what we think of as geometrical truths7 •
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In curved space, the sum of the angles of any triangle is now always greater than *+-.
n a sphere, there are no straight lines. !s soon as you start to draw a straight line, it curves on the sphere.
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In curved space, the shortest distance between any two points (called a geodesic) is not uni3ue. 8or e"ample, there are many geodesics between the north and south poles of the Earth (lines of longitude) that are not parallel since they intersect at the poles.
In curved space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.
)$ *yper%olic Geometry +also called saddle #eometry or Lo%ac"evsian #eometry,: ! non%Euclidean geometry using as its parallel postulate any statement e3uivalent to the following
If l is any line and P is any point not on l , then there e"ists at least two lines through P that are parallel to l .
9obachevskian 2eometry is named for the 5ussian mathematician, :icholas 9obachevsky, who, like 5iemann, furthered the studies of non%Euclidean 2eometry.
;yperbolic 2eometry is the study of a saddle s"aped space. 'onsider what would happen if instead of working on the Euclidean flat piece of paper, you work on a curved surface shaped like the outer surface of a saddle or a $ringle0s potato chip.
#nlike 5iemannian 2eometry, it is more difficult to see practical applications of ;yperbolic 2eometry. ;yperbolic geometry does, however, have applications to certain areas of science such as the orbit prediction of ob
gradational fields, space travel and astronomy. Einstein stated that space is curved and his general theory of relativity uses hyperbolic geometry.
1hat effect does working on a saddle shaped surface have on what we think of as geometrical truths7
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In hyperbolic geometry, the sum of the angles of a triangle is less than *+-.
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In hyperbolic geometry, triangles with the same angles have the same areas.
There are no similar triangles in hyperbolic geometry.
In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.
9ines can be drawn in hyperbolic space that are parallel (do not intersect). !ctually, many lines can be drawn parallel to a given line through a given point.
2raphically speaking, the hyperbolic saddle shape is called ahyperbolic paraboloid , as seen at the right.
It has been said that some of the works of artist =. '. Escher illustrate hyperbolic geometry. In his work Circle Limit III (follow the link below), the effect of a hyperbolic space0s negative curve on the sum of the angles in a triangle can be seen.
Escher0s print illustrates a model devised by 8rench mathematician ;enri $oincare for visualizing the theorems of hyperbolic geometry, the orthogonal circle. =. '. Escher web site httpwww.mcescher.com 'hoose 2alleries 5ecognition and 6uccess *4>>%*4? 'ircle 9imit III
Ans!er t"e follo!in# -uestions dealin# !it" lines and planes$ '$
In this rectangular sided bo", which set of sides lie in the same plane7
C"oose one:
)$ 1hen two planes intersect, two lines are formed. T5#E or 8!96E7 C"oose one: T5#E
8!96E
.$
1hich of the following statements is T5#E7
C"oose one: $lanes l and q are parallel planes
$lanes l and q intersect in line $oint P is in plane l.
.
/$ If two lines intersect, only one plane contains both the lines. T5#E or 8!96E7
C"oose one: T5#E
8!96E
0$
8or this rectangular solid, plane G!and E"C are @@@@@. C"oose one: perpendicular
parallel
neither
1$ 1hich of the following statements is true7 C"oose one:
9ine
lies in plane l .
The intersection of line
and
plane l is point !. $lane l is perpendicular to line
.
2$ If two points lie in a plane, the line
C"oose one: T5#E
8!96E
3$
8or this rectangular solid, which plane(s) contain # and are parallel to plane "EG7 C"oose one: planes #$! and G$#.
planes #C! and "C!.
only plane #$!.
Beach Ball Investigation for Non-Euclidean Geometry Topic Index | Geometry Index | Regents Exam Prep enter
This activity can be accomplished by groups of students or the activity can be performed as a demonstration in front of the class by the teacher andor student volunteers.
Materials: beach balls (or other larger balls) (could be done with balloons with some care) string, protractors, rulers or yardsticks Note: The ball can be marked by the teacher with a permanent marker before the activity begins. This will allow the ball to be used with several different groupsclasses of students. =ark the poles. =ark two other points on the ball allowing for ade3uate distance between the points. =ark the vertices of several different sized triangles on the ball and
draw the triangles using great circles to form the sides. 9abel all of your points for easy reference for students0 answers. The shortest distance between two points on a sphere is along the arc of the great circle
Student Tass: *. #sing the string, determine the length of the #reat circle of the spherical ball. $ull the string tight to the ball between the two poles, to appro"imate a geodesic. 5ecord this length. . 8ind the distance between the two designated, but non%connected, points on the ball. (These points will not be the poles.) 5ecord this length. Is the geodesic you used for this length uni3ue, or are other geodesics possible for this measurement7 /. 9ocate the vertices of each of the triangles on the ball. #sing a great circle as a geodesic, find the lengths of the sides of the triangles. 5ecord the lengths for all of the triangles. A. To the best of your ability, use the protractor to measure the angles in each of the triangles. 5ecord the measurements for each triangle. >. =ake a concluding statement about the relationship between the angels of a triangle on a sphere. B. ! discussion of Euclidean geometry versus non%Euclidean geometry would follow.
Basic onstructions Topic Index | Geometry Index | Regents Exam Prep enter
In geometry, constructions utilize only two tools % the straightedge (an unmarked ruler) and the compass. :ever draw freehand when doing a constructionC
T"e Compass: 'ompasses come in a variety of styles. &ecome familiar with the compass you will be using before beginning your constructions.
6uggestions for working with a compass •
$lace several pieces of paper under your worksheet to allow the compass point to remain stable.
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;old the compass lightly and allow the wrist to remain fle"ible.
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If you cannot manage to move your wrist when drawing circles, try rotating the paper under the compass.
T"e Strai#"ted#e: ! straightedge is generally a clear plastic tool devoid of markings. It most often appears in the shape of a triangle. ! portion of a straightedge is visible in the lower left corner of the picture on the left below. If you do not have a straightedge, a ruler may be used. Dust remember to completely ignore the markings on the ruler.
4asic Geometrical Constructions: The basic constructions used in geometry include
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copy a line segment
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copy an angle
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bisect a line segment
Each construction is developed separately in the following web pages. 6everal videos showing the actual constructions are included.
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bisect an angle
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construct perpendicular lines
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construct parallel lines
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construct isosceles triangle
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construct e3uilateral triangle
1hile the constructions listed above are considered the basic geometrical constructions, you should also be able to construct situations that re3uire the use of these constructions.
opy a !ine "egment and an #ngle Topic Index | Geometry Index | Regents Exam Prep enter
ATTENTION 5ideo Users:
These videos re3uire that you have available a means of displaying video such as 1indows =edia $layer, 5eal $layer, uickTime, etc. Fideo files are lengthy and may take some time to load depending upon your connection. $lease be patient. 1hen the video is loading for the first time, you may e"perience some choppy sound and movement. !llow the video to finish loading and then play again for a smooth delivery.
(emem%er && use your compass and strai#"t ed#e only6
! reference line is a line upon which you produce copies of e"isting figures.
Copy a line se#ment Fideo of 'opy 9ine 6egment Given: (9ine segment) Tas: To construct a line segment congruent to (line segment)
.
7irections:
'$ If a reference line does not already e"ist, draw a reference line with your straightedge upon which you will make your construction. $lace a starting point on the reference line.
)$ $lace the point of the compass on point $. .$ 6tretch the compass so that the pencil is e"actly on !. /$ 1ithout changing the span of the compass, place the compass point on the starting point on the reference line and swing the pencil so that it crosses the reference line. 9abel your copy.
Gour copy and (line segment)
are congruent. 'ongruent means e3ual in length.
E"planation of construction The two line segments are the same length, therefore they are congruent.
Copy an an#le Fideo of 'opy an !ngle Given: Tas: To construct an angle congruent to . 7irections:
'$ If a reference line does not already e"ist, draw a reference line with your straightedge upon which you will make your construction. $lace a starting point on the reference line. )$ $lace the point of the compass on the verte" of (point $). .$ 6tretch the compass to any length so long as it stays : the angle. /$ 6wing an arc with the pencil that crosses both sides of . 0$ 1ithout changing the span of the compass, place the compass point on the starting point of the reference line and swing an arc that will intersect the reference line and go above the reference line.
1$ 2o back to and measure the width (span) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. 2$ 1ith this width, place the compass point on the reference line where your new arc crosses the reference line and mark off this width on your new arc. 3$ 'onnect this new intersection point to the starting point on the reference line.
Gour new angle is congruent to
.
E"planation of construction 1hen this construction is finished, draw a line segment connecting where the arcs cross the sides of the angles. Gou now have two triangles that have / sets of congruent (e3ual) sides. 666 is sufficient to prove triangles congruent. 6ince the triangles are congruent, any leftover corresponding parts are also congruent % thus, the angle on the reference line and are congruent.
Bisect a !ine "egment and an #ngle Topic Index | Geometry Index | Regents Exam Prep enter
ATTENTION 5ideo Users:
These videos re3uire that you have available a means of displaying video such as 1indows =edia $layer, 5eal $layer, uickTime, etc. Fideo files are lengthy and may take some time to load depending upon your connection. $lease be patient. 1hen the video is loading for the first time, you may e"perience some choppy sound and movement. !llow the video to finish loading and then play again for a smooth delivery.
(emem%er && use your compass and strai#"t ed#e only6
4isect & cut into t!o con#ruent +e-ual, pieces$
4isect a line se#ment +Also no! as Construct a Perpendicular 4isector of a se#ment,
Fideo of &isect a 6egment
Given: (9ine segment) Tas: &isect
.
7irections:
'$ $lace your compass point on $ and stretch the compass =5E T;!: half way to point !, but not beyond !. )$ 1ith this length, swing a large arc that will go &T; above and
below
.
(If you do not wish to make one large continuous arc, you may simply place one small arc above
and one small arc below
.)
.$ 1ithout changing the span on the compass, place the compass point on ! and swing the arc again. The two arcs you have created should intersect.
/$ 1ith your straightedge, connect the two points of intersection. 0$ This new straight line bisects
. 9abel the point where the new line and
cross as C .
has now been bisected and $C % C!. (It could also be said that the segments are congruent,
.)
(It may be advantageous to instruct students in the use of the Hlarge arc methodH because it creates a HcrayfishH looking creature which students easily remember and which reinforces the circle concept needed in the e"planation of the construction.)
E"planation of construction To understand the e"planation you will need to label the point of intersection of the arcs above segment as # and below segment as E. raw segments , , and . !ll four of these segments are of the same length since they are radii of two congruent circles. =ore specifically, #$ % #!and E$ % E!. :ow, remember a locus theorem The locus of points e3uidistant from two points, is the perpendicular bisector of the line segment determined by the two points. ;ence, is the perpendicular bisector of . The fact that the bisector is also perpendicular to the segment is actually =5E than we needed for a simple HbisectH construction. Isn0t this greatC 8ree stuffCCC
4isect an an#le Fideo of &isect an !ngle
Given: Tas: &isect
.
7irections: '$ $lace the point of the compass on the verte" of (point $). )$ 6tretch the compass to any length so long as it stays : the angle. .$ 6wing an arc so the pencil crosses both sides of . This will create two intersection points with the sides (rays) of the angle. /$ $lace the compass point on one of these new intersection points on the sides of .
If needed, stretch your compass to a sufficient length to place your pencil well into the interior of the angle. 6tay between the sides (rays) of the angle. $lace an arc in this interior % you do not need to cross the sides of the angle. 0$ 8it"out c"an#in# t"e !idt" of t"e compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Gour two small arcs in the interior of the angle should be crossing. 1$ 'onnect the point where the two small arcs cross to the verte" $ of the angle. Gou have now created two new angles that are of e3ual measure (and are each * the measure of .) E"planation of construction To understand the e"planation, some additional labeling will be needed. 9abel the point where the arc crosses side
as #. 9abel the point
where the arc crosses side as E . !nd label the intersection of the two small arcs in the interior as " . raw segments and . &y the construction, $# % $E (radii of same circle) and 8 J E8 (arcs of e3ual length). f course $" % $" . !ll of these sets of e3ual length segments are also congruent. 1e have congruent triangles by 666. 6ince the triangles are congruent, any of their leftover corresponding parts are congruent which makes e3ual (or congruent) to .
Parallel - through a point Topic Index | Geometry Index | Regents Exam Prep enter
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These videos re3uire that you have available a means of displaying video such as 1indows =edia $layer, 5eal $layer, uickTime, etc. Fideo files are lengthy and may take some time to load depending upon your connection. $lease be patient. 1hen the video is loading for the first time, you may e"perience some choppy sound and movement. !llow the video to finish loading and then play again for a smooth delivery.
(emem%er && use your compass and strai#"t ed#e only6
Parallel %through a point
Fideo of $arallel through a point
Given: $oint P is off a given line Tas: 'onstruct a line through $ parallel to the given line. 7irections:
'. 1ith your straightedge, draw a transversal through point P . This is simply a straight line which runs through P and intersects the given line. )$ #sing your knowledge of the construction '$G !: !:29E, construct a copy of the angle formed by the transversal and the given line such that the cop y is located #$ at point P . The verte" of your copied angle will be point $.
.$ 1hen the copy of the angle is complete, you will have two parallel lines.
This new line is parallel to the given line. E"planation of construction 6ince we used the construction to copy an angle, we now have two angles of e3ual measure in our diagram. In relation to parallel lines, these two e3ual angles are positioned in such a manner that they are called corresponding angles. ! theorem relating to parallel lines tells us that if two lines are cut by a transversal and the corresponding angles are congruent (e3ual), then the lines are parallel.
Perpendiculars - from a point on the line - from a point off the line Topic Index | Geometry Index | Regents Exam Prep enter
ATTENTION 5ideo Users:
These videos re3uire that you have available a means of displaying video such as 1indows =edia $layer, 5eal $layer, uickTime, etc. Fideo files are lengthy and may take some time to load depending upon your connection. $lease be patient. 1hen the video is loading for the first time, you may e"perience some choppy sound and movement. !llow the video to finish loading and then play again for a smooth delivery.
(emem%er && use your compass and strai#"t ed#e only6
Perpendicular & lines +or se#ments, !"ic" meet to form ri#"t an#les$
Perpendicular from a point ON a line
Fideo of $erpendicular n 9ine
Given: $oint P is on a given line Tas: 'onstruct a line through P perpendicular to the given line. 7irections: '$ $lace your compass point on P and sweep an arc of any size that crosses the line twice (below the line). Gou will be creating (at least) a semicircle. (!ctually, you may draw this arc above 5 below the line.) )$ 6T5ET'; T;E '=$!66 9!52E5CC .$ $lace the compass point where the arc crossed the line on one side and make a small arc below the line. (The small arc could be above the line if you prefer.)
/$ 1ithout changing the span on the compass, place the compass point where the arc crossed the line on the T;E5 side and make another arc. Gour two small arcs should be crossing. 0$ 1ith your straightedge, connect the intersection of the two small arcs to point P .
This new line is perpendicular to the given line.
E"planation of construction 5emember the construction for bisect an angle7 In this construction, you have bisected the straight angle P . 6ince a straight angle contains *+ degrees, you have
Perpendicular from a point off a line.
Fideo of $erpendicular ff 9ine
Given: $oint P is off a given line Tas: 'onstruct a line through P perpendicular to the given line.
7irections: '$ $lace your compass point on P and sweep an arc of any size that crosses the line twice. )$ $lace the compass point where the arc crossed the line on one side and make an arc : T;E $$6ITE 6IE 8 T;E 9I:E. .$ 1ithout changing the span on the compass, place the compass point where the arc crossed the line on the T;E5 side and make another arc. Gour two new arcs should be crossing on the opposite side of the line.
/$ 1ith your straightedge, connect the intersection of the two new arcs to point P .
This new line is perpendicular to the given line. E"planation of construction To understand the e"planation, some additional labeling will be needed. 9abel the point where the arc crosses the line as points C and #. 9abel the intersection of the new arcs on the opposite side as point E. raw segments , , , and . &y the construction, PC % P# and EC % E#. :ow, remember a locus theorem The locus of points e3uidistant from two points (C and #), is the perpendicular bisector of the line segment determined by the two points. ;ence,
is the perpendicular bisector of
.
The fact that we created a bisector, as well as a perpendicular, is actually =5E than we needed % we only needed to create a perpendicular. Gea, free stuffCCC
onstructions$ Isosceles and E%uilateral Triangles Topic Index | Geometry Index | Regents Exam Prep enter
irections for constructing isosceles and e3uilateral triangles
Construct an Isosceles Trian#le Usin# Given Se#ment Len#t"s: 1hen constructing an isosceles triangle, you may be given pre%determined segment lengths to use for the triangle (such as in this e"ample), or you may be allowed to determine your own segment lengths. Either way, the construction process will be the same.
'onstruct an isosceles triangle whose legs and base are of the pre% determined lengths given. 'onstruct the new triangle on the reference line.
#sing your compass, measure the length of the given HbaseH.
o not change the size #sing your compass, measure 1ithout changing the Gou now have three of the compass. $lace the length of the given HlegH. size of the compass, points which will define your compass point on $lace the compass point where move the compass point the isosceles triangle. the reference line point the previous arc crosses the to the point on the and scribe a small arc reference line and scribe reference line. 6cribe which will cross the another arc above the reference an arc above the line line. line such that it intersects with the previous arc.
Construct an E-uilateral Trian#le Usin# a Given Se#ment Len#t":
1hen constructing an e3uilateral triangle, you may be given a pre%determined segment length to use for the triangle (such as in this e"ample), or you may be allowed to determine your own segment length. Either way, the construction process will be the same.
'onstruct an e3uilateral triangle whose sides are of given length HaH. 'onstruct the new triangle on the reference line.
#sing your o not change the size of compass, the compass. $lace your measure the compass point on the length of the reference line point and given segment, scribe an arc which will HaH. cross the line and will rise above the line.
o not change the size Gou now have three of the compass. $lace points which will define the compass point the e3uilateral triangle. where the arc crosses the reference line and scribe another arc which crosses the previous arc.
Alternate 9et"od for Constructin# an E-uilateral Trian#le: !n e3uilateral triangle can be easily constructed from a circle. The secret to this method is to remember to keep the compass set at the same length as the radius of the original circle.
raw a circle and place a point on the circle. o not change the size of the compass.
1ith the compass still set at the same size as the radius of the circle, place the compass point on the point on the circle and mark off a small arc on the circle. :ow, move the compass point to this new arc and mark off another arc. 'ontinue around the circle.
Gou now have a circle with si" e3ually divided sections on its circumference.
'onnect every other point on the circle to form the e3uilateral triangle.
onstruction #ctivities Topic Index | Geometry Index | Regents Exam Prep enter
*. raw two 3uadrilaterals of about the same shape as the ones shown below.
#sing your straightedge and compass, bisect each side of each figure. Doin the midpoints of the four sides of each figure in order, so that two new 3uadrilaterals are formed. 1hat do you notice7
. raw a line segment that is several inches long.
#sing your straightedge and compass, divide the segment into four e3ual parts. Into what other number of e3ual parts, less than ten, can a line segment be easily divided7
/.
raw two triangles of the same shape as shown below. #sing your straightedge and compass, bisect each of the three angles of each figure. 1hat do you notice7
A. #sing what you know about constructions, can you figure out a way to construct the following items7 the altitudes of a scalene triangle the altitudes of an obtuse triangle the medians of a scalene triangle the medians of an obtuse triangle a s3uare
>. 'onstruct an angle of /-.
Dustify your construction.
B. 2iven segments of length a and b, construct a rectangle that has a verte" at $ on the given reference line. Dustify why your construction is correct.
?. 2iven the following figure, construct a parallelogram having sides and K $!C . Dustify why your construction is correct.
+. 'onstruct an e3uilateral triangle with sides of length a.
and
Polyhedra Topic Index | Geometry Index | Regents Exam Prep enter
Sin#ular: poly"edron
Plural: poly"edra
! poly"edron is a three%dimensional solid figure in which each side is a flat surface. These flat surfaces are polygons and are
The common polyhedron are pyramids and prisms$
pyramid
prism
! polyhedron is called re#ular if the faces are congruent, regular polygons and the same number of faces meet at each verte". There are a total of five such conve" regular polyhedra called the Platonic solids$
tetrahedron
octahedron
icosahedron
he"ahedron
dodecahedron
Eulers Poly"edron T"eorem: Euler discovered that the number of faces (flat surfaces) plus the number of vertices (corner points) of a polyhedron e3uals the number of edges of the polyhedron plus .
;<5=E<)
Non&Poly"edra The following solids are not polyhedra since a part or all of the figure is curved.
'ylinder
'one
6phere
Torus
! torus is a Htube shapeH. E"amples include an inner tube, a doughnut, a tire and a bagel. 6mallr is the radius of the tube and capital & is the distance from the centre of the torus to the center of the tube.
1hile the torus has a hole in the center, the 6urface !rea the Folume
Prisms Topic Index | Geometry Index | Regents Exam Prep enter
Prisms are three%dimensional closed surfaces.
! prism has two parallel faces, called bases, that are con#ruent polygons. The lateral faces are rectangles in a right prism, or parallelograms in an obli3ue prism. In a right prism, the
$risms are also called polyhedra since their faces are polygons. ! regular prism is a cube.
Right Rectangular Prism
&'li%ue Triangular
Prism
Parallelepiped ! prism which has a parallelogram as its base is called a parallelepiped. It is a polyhedron with B faces which are all parallelograms.
The edges of the prism where the lateral faces intersect are called its lateral ed#es$ The lateral edges in a prism arecon#ruent and parallel$ !ateral edges$ There are 5 congruent and parallel lateral edges in this prism.
The volume of a prism is the product of the base area times the height of the prism.
V = Bh +5olume of a prism: B = %ase area> h = "ei#"t,
h = height(altitude) between bases B = area of the base
The surface area of a prism is the sum of the areas of the bases plus the areas of the lateral faces. This simply means the sum of the areas of all faces.
The surface area, S , of a right prism can be found using the formula S J B L ph. B J area of base, p J perimeter of base, h J height.
! net is a two%dimensional figure that can be cut out and folded up to make a three%dimensional solid.
'ote( ! cross section of a geometric solid is the intersection of a plane and the solid.
A prism "as t"e same cross section +parallel to t"e %ase, all alon# its len#t" 6 6hown here are the cross sections (in the same plane) of two prisms of e3ual height. The cross section slices are indicated in red and are parallel to the bases. If the areas of these two cross section slices are e3ual, the prisms will be e3ual in volume.
6eventeenth century mathematician, &onaventura 'avalieri, generalized this concept for solids.
Cavalieris Principle:
If, in two solids of e3ual height, the cross sections made by planes parallel to and at the same distance from their respective bases are always e3ual, then the volumes of the two solids are e3ual. 8or !lgebra * you should know a generalized statement of this principle
?T!o prisms !ill "ave e-ual volumes if t"eir %ases "ave e-ual area and t"eir altitudes +"ei#"ts, are e-ual$?
(eflective Prisms In the study of optics, prisms are used to reflect light, such as occurs in binoculars. $risms are also used to disperse light, or break light into its spectral colors of the rainbow. The most commonly used optic prism is a triangular prism, which has a triangular base and rectangular sides.
Pyramids
Topic Index | Geometry Index | Regents Exam Prep enter
Pyramid
Pyramids are three%dimensional closed surfaces.
The one %ase of the pyramid is a polygon and the lateral faces are al!ays trian#les with a common verte". The verte" of a pyramid (the point, or ape") is not in the same plane as the base. $yramids are also called polyhedra since their faces are polygons.
1e will be working with regular pyramids unless otherwise stated.
The most common pyramids are re#ular pyramids. ! regular pyramid has a regular polygon for a base and its height meets the base at its center. The slant "ei#"t is the height (altitude) of each lateral face.
In a re#ular pyramid> the lateral edges are congruent. 6ince the base is a regular polygon, whose sides are all congruent, we know that the lateral faces of a regular pyramid are congruent isosceles triangles.
$yramids are named for the shape of their base.
Triangular pyramid
Square pyramid
The volume of a pyramid is one%third the product of the base area times the height of the pyramid.
+5olume of a pyramid: B = %ase area> h = "ei#"t,
h = height (altitude) from vertex to base B = area of base
The surface area of a pyramid is the sum of the area of the base plus the areas of the lateral faces. This simply means the sum of the areas of all faces.
The surface area, S , of a regular pyramid can be
! net is a two%dimensional figure
found using the formula
that can be cut out and folded up to make a three%dimensional solid.
.
B J area of base, p J perimeter of base, s J slant height.
T"e Great Pyramid of E#ypt The 2reat $yramid of Mhufu, at 2iza, Egypt, is ?>* feet long on each side at the base, is A> feet high, and is composed of appro"imately million blocks of stone, each weighing more than tons. The ma"imum error between side lengths is less than .*N. The sloping angle of its sides is >*->*0. Each side is oriented with the compass points of north, south, east, and west. Each cross section of the pyramid (parallel to the base) is a s3uare.
rtist!s rendering of the pyramids.
#ntil the *4th century, the 2reat $yramid at 2iza was the tallest building in the world. !t over A> years in age, it is the only one of the famous 6even 1onders of the !ncient 1orld that remains standing. !ccording to the 2reek historian ;erodotus, the 2reat $yramid was built as a tomb for the $haraoh Mhufu.
ylinders Topic Index | Geometry Index | Regents Exam Prep enter
Cylinder
Cylinders are three%dimensional closed surfaces.
In general use, the term cylinder refers refers to a right circular cylinder with its ends closed to form two circular surfaces, that lie in parallel planes.
h J height (altitude) J radius r J
1e will be working with right circular cylinders unless otherwise stated.
'ylinders are not called polyhedra since their faces are not polygons. In many ways, however, however, a cylinder is similar similar to a prism. ! cylinder has parallel con#ruent %ases, as does a prism, but the cylinder0s bases are circles rather than polygons.
The volume of a cylinder can be calculated in the same manner as the volume of a prism the volume is the product of the base area times the height of the cylinder, V = Bh. 6ince the base in a cylinder is a circle, the formula for the area of a circle can be substituted into the volume formula for B
+5olume of a cylinder: r = = radius of %ase> h = "ei#"t,
two%dimensional ! net is a two%dimensional
The surface area (of a closed cylinder) is a
figure that can be cut out and folded up to make a three% dimensional solid.
combination of the lateral area and the area of each of the bases. 1hen disassembled, the surface of a cylinder becomes two circular bases and and a rectangular surface
(lateral surface), as seen in the net at the left. :ote that the length of the rectangular surface is the same same as the circumference circumference of the the base. 5emember that that the area of a rectangle is length times width. The lateral area (rectangle) J height O circumference of the base. The base area J area of a circle (remember there are two bases)
9ateral J any face or surface that is not a base.
+Total Surface Area of a Closed Cylinder, ...which can also be factored and written as
1hen working with surface areas of cylinders, read the 3uestions carefully.
(ill the surface area include 'oth of the 'ases)
(ill the surface area include only one of the 'ases)
(ill the surface area include neither of the 'ases)
The lateral area only*
ones Topic Index | Geometry Index | Regents Exam Prep enter
Cone
Cones are three%dimensional closed surfaces.
In general use, the term cone refers cone refers to a right circular cone with its end end closed to form a circular base surface. The verte" of the cone (the point) is not in the same plane as the base. h J height (altitude) J radius r J s J slant height
'ones are not called polyhedra since their faces are not polygons. In many ways, however, however, a cone is similar similar to a pyramid. ! cone0s base is simply a circle rather than a polygon as seen in the pyramid.
1e will be working with right circular cones unless otherwise stated.
The volume of a cone can be calculated in the same manner as a s the volume of a pyramid the volume is one%third one%third the product of the base area times times the height of the cone, 6ince the base of a cone is a circle, the formula for the area of a circle can be substituted into the volume formula for ! for !
+5olume of a cone: r = = radius> h = "ei#"t,
! net is a two%dimensional figure that can
The surface area (of a closed cone) is a combination
be cut out and folded folded up to make make a three% dimensional solid.
of the lateral area and the area of the base. 1hen cut along along the slant side and laid flat, the surface of a cone becomes one circular base and the sector of a circle (lateral surface),
as shown in the net at the left. :ote that the length of the arc in the sector is the same as the circumference of the small circular base. &y using a proportion, the area of the sector (lateral area) will be
(measurements pertain to the larger net figure, the circle containing the sector)
9ateral J any face or surface that is not a base.
In a right circular cone, the slant height, s, can be found using the $ythagorean Theorem
Note: The formula for the area of the sector (lateral area), , is e3ual to one half the product of the slant height and the circumference of the base.
(arc length of the sector e3uals the circumference of the smaller base circle)
(the radius of the smaller base is r, while the radius of the larger sector is s)
The lateral area (sector) J The base area J area of a circle
+Total Surface Area of a Closed Cone,
1hen working with surface areas of cones, read the 3uestions carefully.
(ill the surface area include the 'ase)
(ill the surface area not include the 'ase)
"pheres Topic Index | Geometry Index | Regents Exam Prep enter
Sp"ere
Sp"eres are three%dimensional closed surfaces.
! sphere is a set of points in three%dimensional space e3uidistant from a point called the center. The radius of the sphere is the distance from the center to the points on the sphere. r J radius
6pheres are not polyhedra. f all shapes, a sphere has the smallest surface area for its volume.
The volume of a sphere is four%thirds times pi times the radius cubed.
+5olume of a sp"ere: r = radius,
'ote( ! cross section of a geometric solid is the intersection of a plane and the solid.
The surface area of a sphere is four times the area of the largest cross%sectional circle (called the great circle).
! #reat circle is the largest circle that can be drawn on a sphere.
6uch a circle will be found when the cross%sectional plane passes through the center of the sphere. The e-uator is an e"amples of a great circle. =eridians (passing through the :orth and 6outh poles) are also great circles. The shortest distance between two points on a sphere is along the arc of the great circle
8"at "appens !"en planes intersect !it" sp"eres@ *. The intersection of a plane and a sphere is a circle.
:o, not that kind of planeC
. If two planes are e3uidistant from the center of a sphere (and intersecting the sphere), the intersected circles are congruent.
! hemisphere is the half sphere formed by a plane intersecting the center of a sphere.
Platonic "olids +Regular "olids, Regular Polyhedra Topic Index | Geometry Index | Regents Exam Prep enter
5egular solids (regular polyhedra, or $latonic solids which were described by $lato) are solid geometric figures, with identical regular polygons (such as s3uares) as their faces, and with the same number of faces meeting at every corner (verte"). Euclid proved that there are only five regular conve" polyhedra. The five $latonic 6olids were thought to represent the five basic elements of the worldP earth, air, fire, water, and the universe.
The Hregular solidsH are important in many aspects of chemistry, crystallography, and mineralogy. The e3uilateral triangle is the simplest regular polygon. $lacing three e3uilateral triangles at a verte" (total angle *+-) will form a tetra"edron (A faces, A vertices). It has the smallest volume for its surface. The tetrahedron represents fire.
$lacing four e3uilateral triangles at each verte" (total angle AQdefP) will form anocta"edron (+ faces, B vertices). The octahedron rotates freely when held by its two opposite vertices and represents air.
$lacing five e3uilateral triangles at each verte" (/-) will form an icosa"edron ( faces and * vertices). It has the largest volume for its surface area. The icosahedron represents water.
The second simplest regular polygon is the s3uare. $lacing three s3uares at each corner (?-) will form a cu%e, or "ea"edron(B faces and + vertices). The he"ahedron, standing firmly on its base, represents the stable earth.
The third simplest regular polygon is the regular pentagon. $lacing three pentagons at each verte" (/A-) will form adodeca"edron (* faces and vertices). The dodecahedron represents to the universe since the twelve zodiac signs correspond to the twelve faces of the dodecahedron. !nimated solids used with permission of creator 5Rdiger !ppel (all rights reserved).
Tid%it of Info:
! soccer ball is composed of a combination of pentagon and he"agon faces. This shape is called a %ucy%all after 5ichard &uckminster 8uller, who invented the geodesic dome. In reality, the soccer ball is not truly a polyhedron since the faces are not really flat. The faces tend to bulge slightly due to the amount of stuffing in the ball and the pliable nature of the leather.
Ans!er t"e follo!in# -uestions dealin# !it" .&dimensional fi#ures$ !nswers use J /.*A*>4B>A, the full calculator entry on the TI%+/L+AL, and are rounded to the nearest tenth unless otherwise stated.
'.
C"oose:
*+ cu. units 4/.> cu. units +.> cu. units AB?.? cu. units This figure is a regular he"agonal prism. 8ind its volume.
"xplanation
).
C"oose:
*.? s3. ft. >. s3. ft. The radius of this sphere is A feet. 8ind the area of the great circle of this sphere to the nearest tenth.
"xplanation
>./ s3. ft. +A. s3. ft.
..
C"oose:
** 8ind the volume of this right circular cone to the nearest cubic foot.
+ BA A?
"xplanation
/.
!t the age of +*, =r. 9uke 5oberts decided to start collecting string. ;e has a spherical ball of string three feet in diameter. n the assumption that one cubic inch of string weighs ./ pounds and the ball was solid string, how much does the ball weigh to thenearest tenth of a pound 7
C"oose:
?/. lbs >+/. lbs ?/.4 lbs AA.4 lbs
"xplanation
0.
! cylinder and a cone each have a radius of / cm. and a height of + cm. 1hat is the ratio of the volume of the cone to the volume of the cylinder7
C"oose:
Fcone Fcylinder J * Fcone Fcylinder J * / Fcone Fcylinder J * Fcone Fcylinder J * *
"xplanation
1.
C"oose:
The figure to the left is a cube. 8ind the number of degrees in angle
A>-
B"xplanation
4-
2.
C"oose:
True 8alse
If this %dimensional net is assembled, it will form the /% dimensional figure shown at the right.
3.
C"oose:
*+ s3. in. A s3. in. /B s3. in. This figure represents a slab of cheese. It is in the form of a right triangular prism. 8ind the least amount of wrapping needed to cover the cheese on all sides.
A+ s3. in.
"xplanation
B.
1hen you blow up a balloon it forms a sphere because it is trying to hold as much air as possible with as small a surface as possible.
C"oose:
*>.+ cu.in. B. cu.in.
;ow much air, to the nearest tenth of a cubic inch, is being held by a spherical balloon with a diameter of * inches7
4A.+ cu.in. ?/+. cu.in.
"xplanation
'.
C"oose:
+ ! regular pyramid is shown at the left. 8ind the volume of the pyramid to the nearest cubic unit.
B? + ?
"xplanation
''.
C"oose:
*>?> 8ind the volume in cubic feet.
A> >B> B>
"xplanation
').
The lateral surface area of a right circular cone, L) , is represented by the e3uation , where r is the radius of the circular base and h is the height of the cone. If the lateral surface area of an ice cream cone is /B.BA s3uare centimeters and its radius is A.?> centimeters, find its height, to the nearest hundredth of a centimeter.
C"oose:
>*.A? cm +.4* cm *>.+B cm *>.*/ cm
"xplanation
'..
! cylinder and a sphere have the same radius and the same height. 1hat is the ratio of the volume of the cylinder to the volume of the sphere7
C"oose:
Fcylinder Fshere J * * Fcylinder Fsphere J *
Fcylinder Fsphere J
/
Fcylinder Fsphere J /
"xplanation
'/.
!t the =etro$le" movie theater, popcorn is served in a bo". !t the 'inema$le" movie theater, popcorn is served in a cylindrical container. !t home, =om serves popcorn in a bowl (hemisphere in shape). &ased upon the given dimensions, where are you getting the most popcorn7 (isregard the thickness of the container.)
C"oose:
=etro$le" 'inema$le" =om0s $lace
"xplanation
'0.
C"oose:
A*.B cu. units *A.? cu.
This figure is a regular he"agonal prism. 8ind its volume.
units *+.A cu. units /.B cu. units
"xplanation
'1$
In the chart below, the dimensions of a cylinder and a cone are doubled and tripled. 8ind the volume of all three cylinders and all three cones. 1hen finished, state a hypothesis as to what happens to the volume of solids when
the sides are doubled and tripled. #ylinder$
r = %& h = '
oubled$ r = & h = *%
Tripled$ r = *+& h = +*
#one$
r = %& h = '
oubled$ r = , h = *%
Tripled$ r = *+& h = +*
-ypothesis
Polygons Topic Index | Geometry Index | Regents Exam Prep enter
! poly#on is a closed figure that is the union of line segments in a plane. ! polygon has three or more sides. ! polygon has the same number of angles as sides.
$olygons can be classified as either conve or concave.
! polygon is conve if no line that contains a side of the polygon contains a point in the interior of the polygon. In a conve polygon, each interior angle measures less than *+ degrees. Concave polygons Hcave%inH to their interiors, creating at least one interior angle greater than *+ degrees (a refle" angle). #nless otherwise stated, we will be discussing conve polygons.
Types of Poly#ons 9isted below are some of the more commonly used polygons. (o not assume that the diagrams under the H2raphicH column are HregularH polygons. o not assume any specific details about the diagrams such as the length of the sides or measures of the angles.) "ides
#ngles
.ertices
/iagonals
Num'er Triangles
Triangle
/
/
/
*
uadrilateral
A
A
A
$entagon
>
>
>
>
/
;e"agon
B
B
B
4
A
Polygon
Graphic
;eptagon or 6eptagon
?
?
?
*A
>
ctagon
+
+
+
B
:onagon or :ovagon
4
4
4
?
?
ecagon
*
*
*
/>
+
odecagon
*
*
*
>A
*
n
n
n
n%gon
%%%
! polygon is e-uilateral if all of its sides are of the same length. ! polygon is e-uian#ular if all of its angles are of e3ual measure. ! re#ular poly#on is a polygon that is both e3uilateral and e3uiangular.
"um of Interior #ngles
(n % )
of a Polygon Topic Index | Geometry Index | Regents Exam Prep enter
Sum of Interior An#les of a Poly#on
= '3+n & ), (where n J number of sides)
Lets investi#ate !"y t"is formula is true$
6tart with verte" A and connect it to all other vertices (it is already connected to & and E by the sides of the figure). Three triangles are formed. The sum of the angles in each triangle contains *+-. The total number of degrees in all three triangles will be / times *+. 'onse3uently, the sum of the interior angles of a pentagon is
/
*+ J >A
:otice that a pentagon has 0 sides, and that . triangles were formed by connecting the vertices. The number of triangles formed will be ) less than the number of sides.
This pattern is constant for all polygons. 5epresenting the number of sides of a polygon as n, the number of triangles formed is +n & ),. 6ince each triangle contains '3D, the sum of the interior angles of a polygon is '3+n & ),.
Usin# t"e ;ormula There are two types of problems that arise when using this formula
*. uestions that ask you to find the number of degrees in the sum of the interior angles of a polygon. . uestions that ask you to find the number of sides of a polygon.
*int: 1hen working with the angle formulas for polygons, be sure to read each 3uestion carefully for clues as to which formula you will need to use to solve the problem. 9ook for the words that describe each kind of formula, such as the words sum, interior, each, e"terior and degrees.
Eample ':
8ind the number of degrees in the sum of the interior angles of an
octagon.
!n octagon has + sides. 6o n J +. #sing the formula from above,'3+n & ), J *+(+ % ) J *+(B) J '3 degrees.
Eample ):
;ow many sides does a polygon have if the sum of its interior angles is ?-7
6ince, the number of degrees is given, set the formula above e3ual to ?-, and solve for n.
'3+n & ), J ? n % J A n=1
6et the formula J ?ivide both sides by *+ !dd to both sides
Each Interior #ngle of a Regular Polygon Topic Index | Geometry Index | Regents Exam Prep enter
5emember that the sum of the interior angles of a polygon is given by the formula
Sum of interior an#les = '3+n & ), where n J the number of sides in the polygon.
! polygon is called a (EGULA( polygon when all of its sides are of the same length and all of its angles are of the same measure. ! re#ular poly#on is both e3uilateral and e3uiangular.
9et0s investigate the regular pentagon seen above. To find the sum of its interior an#les> substitute n = 0 into the formula '3+n & ), and get '3+0 & ), = '3+., = 0/D 6ince the pentagon is a re#ular pentagon, the measure of each interior angle will be the same. To find the size of each angle, divide the sum, >AS, by the number of angles in the
pentagon. (which is the same as the number of sides).
0/D
0 = '3D
There are *+- in eac" interior angle of a regular pentagon.
T"is process can %e #eneralied into a formula for findin# eac" interior an#le of a (EGULA( poly#on $$$ Eac" interior an#le of a ?re#ular? poly#on =
where n J the number of sides in the polygon.
&e carefulCCC If a polygon is :T 5E2#9!5 (such as the one seen at the right), you cannot use this formula. If the angles of a polygon 7O NOT all have the same measure, then you cannot find the measure of any one of them
Eamples EACH appears in the question, you +ill &ead these questions carefully* If the +ord most li-ely need the formula for each interior angle to sole the problem.
'$ 8ind the number of degrees in eac" interior angle of a regular dodecagon. It is a regular regular polygon, polygon, so we can use the formula. In a dodecagon, n J *.
polygon measures */>-. )$ Eac" interior angle of a regular polygon
;ow many sides does the the polygon
have 7 •
8irst, set the formula (for each interior angle) e3ual to the number of degrees given.
•
'ross multiply.
•
=ultiply *+ by (n ( n % ).
•
•
6ubtract */>n */>n from both sides of the e3uation. ivide both sides of the e3uation by A>.
Exterior #ngle Topic To pic Index Inde x | Geometry Index | Regents Exam Prep enter
!n eterior an#le of a polygon is an angle that forms a linear pair with
one of the angles of the polygon.
Two Two e"terior angles can be formed at each verte" of a polygon. The e"terior angle is formed formed by one side of the polygon and the e"tension e"tension of the ad
Note: 1hile it is possible to draw T1 (e3ual) e"terior angles at each verte" of a polygon, the sum of the e"terior angles formula uses only :E e"terior angle at each verte".
0ormula$ Sum eterior Sum eterior an#les of any any poly#on poly#on = .1D (using one e"terior angle at a verte")
8inding the sum sum of of the e"terior angles of a polygon is simple. simple. :o matter what what type of polygon you have, the the sum of the e"terior angles is is AL8AFS e3ual to /B/B-..
If you are working with a regular polygon, you can determine the size of EAC* e"terior angle by simply dividing the sum, /B, by the number of angles. 5emember, the formula below will :9G work in a regular polygon. polygon.
0ormula$ Eac" eterior Eac" eterior an#le +re#ular +re#ular poly#on, poly#on, =
Eamples the sum of the e"terior angles of '$ 8ind Ans!er: a) a pe pentagon b) a decagon c) a *> *> si sided po polygon d) a ? sided po polygon
.1 Ans!er: .1 Ans!er: .1 Ans!er: .1
)$ 8ind the measure of eac" e"terior angle of a regular he"agon. A "ea#on "as 1 sides> so n = 1 Su%stitute in t"e formula$
.$ The measure of eac" e"terior angle of a regular polygon is A>-.
;ow many sides does the
polygon have 7
Set t"e formula e-ual to /0 $ Cross multiply and solve for n$
Types of "entences Topic Index | Geometry Index | Regents Exam Prep enter
ne of the goals of studying mathematics is to develop the ability to think critically. The study of critical thinking, or reasoning, is called lo#ic$
!ll reasoning is based on the ways we put sentences together. 9et0s start our e"amination of logic by defining what types of sentences we will be using. ! mathematical sentence is one in which a fact or complete idea is e"pressed. &ecause a mathematical sentence states a fact, many of them can be
•
H!n isosceles triangle has two congruent sides.H is a true mathematical sentence.
•
H* L A J *>H is a false mathematical sentence.
•
Hid you get that one right7H is :T a mathematical sentence % it is a question.
•
H!ll trianglesH is :T a mathematical sentence % it is a phrase.
There are two types of mathematical sentences !n open sentence is a sentence which contains a variable. •
H x L J +H is an open sentence %% the variable is H x.H
•
HIt is my favorite color.H is an open sentence%% the variable is HIt.H
•
The truth value of theses sentences depends upon the value replacing the variable.
! closed sentence, or statement, is a mathematical sentence which can be
H2arfield is a cartoon character.H is a true closed sentence, or statement.
•
H! pentagon has e"actly A sides.H is a false closed sentence, or statement.
! compound sentence is formed when two or more thoughts are connected in one sentence.1ords such as and, or, if***then and if and only if allow for the formation of compound sentences, or statements. :otice that more than one truth value is involved in working with a compound sentence. •
HToday is a vacation day and I sleep late.H
•
HGou can call me at * o0clock or you can call me at o0clock.H
•
H If you are going to the beach, then you should take your sunscreen.H
•
H! triangle is isosceles if and only if it has two congruent sides.H
6entences, or statements, that have the same truth value are said to be
lo#ically e-uivalent$ (He3uivalentH means He3ualH)
Negation -- N&T-"imple "tatements Topic Index | Geometry Index | Regents Exam Prep enter
In logic, a negation of a simple statement (one logical value) can usually be formed by placing the word HnotH into the original statement. The negation will always have the opposite truth value of the original statement.
#nder negation, what was T5#E, will become 8!96E % or % what was 8!96E, will become T5#E.
Eamples of simple ne#ations: '$ riginal 6tatement
H*> L e3uals />.H (is true) :egation H*> L does not e3ual />.H (is false)
)$ H! dog is a cat.H
is a false statement. H! dog is not a cat.H is a true statement. HIt is not true that a dog is a cat.H is a true statement. HIt is not the case that it is not true that a dog is not a cat. H is a true statement. :otice that there are different ways of inserting the concept of HnotH into a statement. 1hile we would not usually speak in a manner similar to the last statement, we must be alert to people who attempt to win arguments by using several negations at the same time to cause confusion.
.$ H! fish has gills.H is a true statement.
H! fish does not have gills.H is a false statement. HIt is not true that a fish does not have gills.H is a true statement$ :otice how using T1 negations, returns the truth value o f the statement to its original value. In plain English, this means that two negations will HundoH one another (or cancel out one another).
/$
riginal statement HDedi masters do not use light sabers.H :egation HDedi masters do not not use light sabers.H &etter :egation HDedi masters do use light sabers.H :otice even though the first negation shows the proper insertion of the word HnotH, the second negation can be more easily read and understood.
=athematicians often use symbols and tables to represent concepts in logic. The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.
Truth table for negation (not) (notice the symbol used for HnotH in the table below)
! truth table is a pictorial representation of all of the possible outcomes of the truth value of a sentence. ! letter such as is used to represent the sentence or statement. T
;
;
T
(E9E94E(:
#nder negation, T5#E becomes 8!96E % or % 8!96E becomes T5#E.
on1unction #N/ Topic Index | Geometry Index | Regents Exam Prep enter
In logic, a con
! con> minutes and one minute J e"actly B seconds.H (8 and T J 8) 6ince the first fact is false, the entire sentence is false. .$ H/ L A J B and all dogs meow.H (8 and 8 J 8) 6ince both facts are false, the entire sentence is false.
=athematicians often use symbols and tables to represent concepts in logic. The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.
Truth table for con
! truth table is a pictorial representation of all of the possible outcomes of the truth value of a T
T
T
T
;
;
;
T
;
;
;
;
compound sentence. 9etters such as and are used to represent the facts (or sentences) within the compound sentence.
(emem%er:
;or a conHunction +and, to %e true> 4OT* facts must %e true$
/is1unction &R Topic Index | Geometry Index | Regents Exam Prep enter
In logic, a dis
! dis
)$ Hne hour J e"actly >> minutes or one minute J e"actly B seconds.H (8 or T J T) 6ince the second fact is true, the entire sentence is true.
.$ H/ L A J B or all dogs meow.H (8 or 8 J 8) 6ince both facts are false, the entire sentence is false.
/$
HThe word cat has / letters or the word dog has four letters.H (T or 8 J T) 6ince the first fact is true, the entire sentence is true.
=athematicians often use symbols and tables to represent concepts in logic. The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.
Truth table for dis
! truth table is a pictorial representation of all of the possible outcomes of the truth value of T
T
T
T
;
T
;
T
T
;
;
;
a compound sentence. 9etters such as
and
are used to represent the facts (or sentences) within the compound sentence.
(emem%er: ;or a disHunction +or, to %e true> EIT*E( or 4OT* facts must %e true$
Biconditional I0 #N/ &N!2 I0 Topic Index | Geometry Index | Regents Exam Prep enter
In logic, a biconditional is a compound statement formed by combining two conditionals under Hand.H &iconditionals
are true when %ot"statements (facts) have the eact same trut" value$ ! biconditional is read as Hsome factU if and only if another factUH and is true when the truth values of both facts are e"actly the same %% &T; T5#E or &T; 8!96E.
4iconditionals are often used to form definitions$ 7efinition: ! triangle is isosceles if and only if the triangle has two congruent (e3ual) sides. The Hif and only if H portion of the definition tells you that the statement is true when either sentence (or fact) is the hypothesis. This means that both of the statements below are true If a triangle is isosceles, then the triangle has two congruent (e3ual) sides. (true)
If a triangle has two congruent (e3ual) sides, then the triangle is isosceles. (true)
=athematicians often use symbols and tables to represent concepts in logic. The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.
Truth table biconditional (if and only if) (notice the symbol used for Hif and only ifH in the table below)
T
T
T
T
;
;
;
T
;
;
;
T
! truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence. 9etters such as and are used to represent the facts (or sentences) within the compound sentence.
I
(E9E94E(: I8 !: :9G I8 is T5#E when both facts are T or both facts are 8..
Truth .alue of &pen "entences
Topic Index | Geometry Index | Regents Exam Prep enter
6ometimes it is difficult to determine the truth value of a sentence. Even though the sentence conveys a complete thought, the sentence may be true for some people and false for others.
;or eample: •
/roccoli tastes awful.
•
The /eatles were an awesome group.
•
0ootball is the most exciting sport to watch.
!n even worse situation is the case where it is impossible to determine the truth value of a sentence due to a lack of information.
;or eample: •
She did her homewor1.
•
x 2 5 = +5
•
3t!s the best movie this year.
6entences that lack information are called Open Sentences6
!n open •
sentence is a sentence which contains a variable.
She did her homewor1. is an open sentence %% the variable is She.
•
x 2 5 = +5 is an open sentence %% the variable is x.
•
3t!s the best movie this year. is an open sentence%% the variable is 3t.
! variable is simply a spot waiting for a value. The values we put into the variable are called thedomain, or replacement set (because they HreplaceH the variable.) The set of values which make the sentence T5#E is called the solution set, or truth set.
Eample: Open sentence: x L > J > 5aria%le: x 7omain:
V*, , , AW
Solution Set:
VW
(the answer which makes the open sentence true)
(numbers you can choose from)
Eample: Open sentence: 6he did her homework.
5aria%le:
6he
7omain:
V6ue, =elissa, Dennifer, 6andy, DoanneW (girls0 names you can choose from)
Solution Set: V6ue, 6andyW (the answers which makes the open sentence true) (Gou would have to know which girls I their homework. In this case, 6ue and 6andy did their homework.)
(emem%er:
pen sentences re3uire that you have additional information to determine whether they are true or false.
ompound "entences Topic Index | Geometry Index | Regents Exam Prep enter
! compound sentence is formed when two or more thoughts are connected in one sentence. The following are e"amples of compound sentences *. +* is divisible by 4 and +* is not prime. . %5 is a multiple of or *4 6 +7 = '. /. If % 2 8 = *7 and 4 2 4 = , then all rectangles are squares.
1hen attempting to determine the truth value of a compound sentence, first determine the truth value of each of the components of the sentence. 9et0s e"amine the e"amples listed above.
*. etermine the truth value of
+* is divisible by 4
+* is divisible by 4 (true) +* is not prime (true)
6ubstitute the truth values for the facts 6implify the conHunction (and )
T and T T
and +*
is not prime.
!nswer The compound sentence (statement) is true.
. etermine the truth value of
%5 is a multiple of
or *4
6 +7 = '.
%5 is a multiple of (true) *4 6 +7 = ' (false) 6ubstitute the truth values for the facts 6implify the disHunction (or )
T or 8 T
!nswer The compound sentence (statement) is true.
/. etermine the truth value of If % 2 8 = *7 and 4 2 4 = , then all rectangles are squares.
% 2 8 = *7 (true) 4 2 4 = (false) all rectangles are squares. (false) 6ubstitute the truth values for the facts 6implify the conHunction (and ) first
if (T and 8) then 8 if 8 then 8
6implify the conditional
T
!nswer The compound sentence (statement) is true.
1hen the truth value of one or more of the components of a compound sentence is unknown, all of the possible truth values must be considered. ! truth table is the easiest way to show all of these possibilities.
'onstruct a truth table for
T
T
T
T
;
;
T
;
;
T
;
T
;
T
;
T
;
T
;
;
;
;
T
T
The truth table tells you that the compound sentence will be false only when p and q are both true. In all other situations, the compound sentence is true.
Negation -- N&T-ompound "tatements and #!! 3 "&4E Topic Index | Geometry Index | Regents Exam Prep enter
1e know that the negation of a true statement will be false, and the negationof a false statement will be true. &ut what happens when we try to
ne#ate a compound statement@ Ne#atin# a ConHunction +and , and a 7isHunction +or ,: The negation of a con
H6noopy wears goggles and scarves.H HIt is not the case that 6noopy wears goggles and scarves.H 1hile by our negation we know that 6noopy does not wear &T; goggles and scarves, we cannot say for sure that he does not wear :E of these items. 1e can only state that he does not wear goggles or he does not wear scarves. ( ) 7isHunction:
HI will paint the room blue or green.H HIt is not the case that I will paint the room blue or green.H If I am not painting the room blue or green, then I am not painting EIT;E5 color. 6o it can be said that HI am not painting the room blueH and HI am not painting the room greenH.
7e9or#ans La!s: +negating AN7 and O( ) (The statements shown are logically e3uivalent.)
:otice that the negation symbol is distributed across the parentheses and the symbols are changed from!: to 5 (or vice versa).
Ne#atin# a Conditional +if ... then,: 5emember 1hen working with a conditional, the statement is only ;ALSE when the hypothesis (HifH) is T(UE and the conclusion (HthenH) is ;ALSE$ HIf 4 L / J *, then 4 is a prime number.H is a 8!96E statement. HIt is not the case that if 4 L / J *, then 4 is a prime number.H is T5#E. H4 L / J * and 4 is not a prime number.H is a T5#E statement.
Ne#ate a Conditional: +negating I; $$$ T*EN) :otice that the statement is re%written as a con
Ne#atin# a 4iconditional +if and only if ,:
5emember 1hen working with a biconditional, the statement is T(UE only when both conditions have the same truth value. H! triangle has only / sides if and only if a s3uare has only A sides.H
... is logically equivalent to ... HIf it is a triangle then it has only / sides and if it is a s3uare then it has only A sides.H To negate a biconditional, we will negate its logically e3uivalent statement by using e=organ0s 9aws and 'onditional :egation. X X X X
Ne#ate a 4iconditional: +negating I; AN7 ONLF I;)
Ne#atin# ALL and SO9 E: ALL Consider( H!99 students are opera singers.H (=eaning that there are : students who arenot opera singers.)
SO9E Consider( H6ome rectangles are s3uares.H (=eaning that there e"ists at
