MANISH KUMAR
MATHEMATICS
CO-ORDINATE GEOMETRY
INTRODUCTION: In this chapter, we shall first define the coordinates of a point in a plane with reference to two mutually perpendicular lines in the same plane. We shall also leans about the plotting of points in the plane (Cartesian plane) which will be used draw the graphs of linear equations in one/two variables in the cartesian plane. CARTESIAN CO-ORDINATE SYSTEM : =
(i)
Cartesian co-ordinate axes : - Let x’ox and y’oy be two mutually perpendicular lines such that x’ox is horizontal and y’oy is vertical line in the same plane and they intersect each other at O. The line x’ox called the x-axis or axis of x the line y’oy is called the y-axis of y and the two lines x’ox and y’oy taken together are called the co ordinate axes or the axes of co-ordinates. The point ‘O’ is called the origin.
(ii)
Quadrants: - The co-ordinate axes x’ox and y’oy divide the plane of graph paper in the four regions xoy, x’oy x’oy, xoy’. These four regions are called the quadrants. The regions xoy, x’oy, x’oy and xoy’ are known as the first, the second, the third and the fourth quadrant respectively.
(iii)
Cartesian co-ordinate of a point :- Let x’ox and y’oy be the co-ordinate axes and let P be any point in the plane. To find the position of P with respect to x’ox and y’oy. WE draw two perpendiculars from P on both coordinate axes. Let PM & PN be perpendiculars on x-axis and y-axis respectively. Draw PM x’ox and PN y’oy.
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The length of the line segment OM is called the x=coordinate or abscissa of point P and the length of the directed line segment ON is called the y-co-ordinate or ordinate of point P. Let OM = 3 & ON = 5, then the x-xoordinate or abscissa of point P is 3 and the y-co-ordinate or ordinate of P is 5 and say that the co-ordinate of P are (3, 5). Thus for a given point P, the abscissa and ordinate are the distance of the point P from y-axis and x-axis respectively. The above system of co-ordinating an ordered pair (3,5) with every point in a plane is called the Rectangular cartesian co-ordinate system.
(iv)
Convention of Signs :- Let x’ox and y’oy be the co-ordinate axes. As discussed earlier that the regions xoy, x’oy, x’oy, and xoy’ are known as the first, the second, the third and the fourth quadrant respectively. The ray ox is taken as positive x-axis, ox’ as negative x-axis, oy as positive y-axis and oy’ as negative y-axis. We find that In
(v)
I
quadrant
x>0
y>0
(+ , +)
II
quadrant
x<0
y>0
(-, + )
III
quadrant
x<0
y<0
( -, - )
IV
quadrant
x>0
y<0
(+, - )
Points on axes :- Point P lies on x axis then clearly the distance of this point from x-axis is zero and therefore the y ordinate of this point is O. If any point lies on x-axis then its y co-ordinate will be zero (x, o). Similarly, if we take a point on y-axis than if distance from y-axis is o and the x-co-ordinate of this point is zero. In general, if any point lies on y-axis then its x-co-ordinate will be zero (o, y). The co-ordinate of the origin are taken as (o, o).
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IIIustrative Examples :Ex.1
Plot the following points on a graph paper. (i) (3,4 )
Sol.
(ii) (-2, 3)
(iii) (-5, -2)
(iv) (4, -3)
Let
be
X’OX and Y’OY
10 9 8 7 6 Y 5 4 3 2 1
the
The given four potted as given
coordinate axes. points may be below :-
0 8
7
6
5
4
- 3 B(-2,3) 2
1
1
2
A(3,4) 3 4
-1 -2 -3 -4 -5 -6 -7 -8 -9
X’
C(5,-2)
5
6
7
8
X’
D(4,-3)
Y’ Ex .2
Sol.
Write the quadrants for the following points :(i) (-2, 3)
(ii) (-5, -2)
(iii) (4, -3)
(iv) (-5, -5)
(i)
Here x is negative & y is positive Point lies in II quadrant.
(ii)
Here both x & y coordinate are negative. Point lies in III quadrant.
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(iii)
Here x is positive & y is negative. Point lies in IV quadrant.
(iv)
Here both x & y co-ordinates are negative Point lies in III quadrant.
(v)
Here both x & y co-ordinates are positive Point lies in I quadrant.
MATHEMATICS
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INTRODUCTION TO EUCLID’S GEOMETRY
INTRODUCTION : The word ‘geometry’ is derived from two Greek words ‘geo’ (meaning ‘earth’) and ‘matron’ (meaning to measure’). Geometry appears to have originated from the need for measuring land.
1. 2.
GEOMETRY : Generally, Geometry may be considered as : The study of size, shape, position and other properties of the objects around us. A mathematical system in which a few basic statements or ideals are agreed to and then used to verify results by logical reasoning. POINT : A point has position only. It has no length, no width and no thickness. A point is represented by a fine dot and is denoted by capital letter P, A, R, S, etc.
L N M O
The adjoining figure shows the points L, M, N and O. A point is generally represented by a dot but the dot is not a point. For example, a locality on a map may be represented by a dot, but the dot is not the locality. LINE :
l
A line has length but no width and no thickness. A light thread or a straight crease, obtained by folding a paper, is closed to a segment of a line. The adjoining figure shows a line passing through two points A and B. So this line can be represented by saying ‘line AB’.
B A
It can also be represented by a small letter. Thus the given figure represented by a small letter. Thus the given figure represented line AB or line . PALNE : If any two points are taken anywhere on a surface and joined by a straight line, then if each and every point of this line lines in the surface, the surface is called a plane or a plane surface. Thus, a plane is a surface such that straight line, passing through any two points of it, lies entirely in it. ASSUMPTIONS : AXIOMS AND POSTULATES In order to prove truths in mathematics (specially, in geometry), Euclid assumed certain properties (statements) as being true without proof. Euclid divided these assumptions into two parts, Axioms and Postulates. An axiom is simply an assumption used in logic and algebra, whereas a postulate was reserved for an assumption in
1.
geometry. But nowadays, these terms are used interchangeably. Some of the Euclid’s Axioms and Postulates are given below : A quantity is equal to, greater than, or less than any other quantity.
2. 3. 4. 5.
When equals are added to equal, their sums are equal. When equals are subtracted from equals, the differences are equal. When equal are multiplied by equals the products are equal. When equals are divided by equals the quotients are equal.
6.
Things equal to the same thing, or to equal things, are equal to one anther.
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7.
The whole (a whole thing) is equal to the sum of all its parts.
8. 9.
Through a given external point, one and only one line can be drawn parallel to a given line. Two non parallel straight lines can intersect in only one point.
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10.
Through a given point, an unlimited number of straight lines can be drawn.
11.
A line segment an be produced to any desired length.
12.
The whole is greater than any of its parts.
13.
Through two given points, it is possible to draw one, and only one, straight line.
14.
When two lines intersect, the vertically opposite angles, formed in pairs, are equal.
15.
The shorted path between two given points is the line segment which joins them.
16.
Through a give point, one and only are perpendicular can be drawn to a given line.
17.
An angle can be bisected by one and only one line.
18.
A line segment can be bisected at one and only one point.
19.
All right angles are equal.
20.
The sum of all angles having a common vertex on a straight line and lying on the same side of the line is 180 0
21.
Things which coincide with one another are equal to one another.
22.
Things which are halves of the same things are equal to one another.
23.
Things which are double of the same things are equal to one another. Out of the many axioms and postulates, we now discuss the following five-postulates given by Euclid.
(A)
FIRST POSTULATE :- A straight line may be drawn from any point to any other point. Let A be a given point and B be some other point. If we draw several lines passing through the point A we sea that only one of these lines passes through the point B also. Similarly, if we draw several lines passing through the point B we see that only one of these lines passes through the point A also. So, we can say a unique line passes through the points A and B. This result in the form of an axiom is as follows :
One and only one line passes through two distinct points. (B)
SECOND POSTULATE :- A terminated line (line segment) can be produced indefinitely. The adjoining figure shows a terminated line (line segment) AB with two end points A and B.
B
(A terminated line)
A According to the second postulate, this terminated line can be extended on either side of it to get a line, as per the figure.
B
(A line)
A (C)
THIRD POSTULATE : A circle can be drawn with any centre and any radius.
(D)
FOURTH POSTULATE : All right angles are equal to one another.
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(E)
FIFTH POSTULATE : If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
1.
The adjoining figure shows a line PQ falling on lines AB and CD such that the sum of the interior angle 1 and 2 is less than 1800 on the left side of PQ. Therefore, lines AB and CD will eventually intersect on the left side of PQ.
2.
The figure given alongside shows a line MN falling on lines AB and CD such that the sum of the interior angles 3 and 4 is less than 1800 on the right side on MN. Therefore, the lines AB and CD will eventually intersect on the right side on MN. PROPOSITIONS OR THEOREMS : After stating his postulated and axioms, Euclid used them to prove (deduce) other results called propositions or theorems. He deduced 456 propositions in a logical chain using his postulates, axioms, etc. In geometry, proposition are accepted as true if they can be reached by deductive method. A Theorem is statement of a geometrical fact which has been proved or needs to be proved. For example “The sum of the angles of a triangle is equal to two right angles”. A corollary is a statement, the truth of which may readily be inferred from a theorem. For example a corollary to the above theorem is :“Each angle of an equiangular triangle is equal to 600”.
Ex.1
Define : (i) Line (ii) Ray (iii) Line segment
(iv) Collinear points
(v) Intersecting lines
(vi) Concurrent lines
(vii) Parallel lines. Sol.
For each definition, draw a suitable diagram:
(i)
Line : A line has length but no width or thickness. A line is unlimited in extent.
B A
It extends in both the direction without end. The given figure shows a line AB. (ii)
Ray : A straight line, generated by a point and moving in the same direction is called a ray. The given figure shows a ray AB which is generated by the point A and is moving in the same direction AB.
(iii)
B
A
Line segment : It is the part of a line whose both the ends are fixed
B
(terminated). The given figure shows a line segment AB. (iv)
Collinear points: Three or more points lying on the same straight line are called collinear points. In the given figure, points A, B and C lie on the same
A
straight line, so the points are collinear.
A
B
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(v)
MATHEMATICS
Intersecting lines : If two lines have a common point, the lines are said to be intersecting lines. In the given figure, line and m have point O common. therefore these are intersecting lines.
(vi)
Concurrent lines : Three or more lines in a plane are said to be concurrent if all of them pass through the same point. In the given figure, four lines are passing through the same point 0, therefore these lines are concurrent lines. The common point (point O in this case) is called the point of concurrency.
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(vii)
Parallel lines : Two lines are said to be parallel to each other if they never intersect, no matter upto what extent are they produced in any direction.
A C
B B D
The given figure shows two parallel lines, AB and CD which will never intersect on producing both of these in any direction and upto any extent. Ex.2
If a point C lies between two points A and B such that AC = BC, then prove that AC =
1 AB. Explain by drawing 2
the figure. Sol.
According to the given statement, the figure will be shown alongside in which the point C lies between two point A and B such that AC = BC. Clearly,
AC + BC = AB
AC + AC = AB
Ex.3
Sol.
2AC = AB =
||
[ AC = BC]
||
B
C
A
1 AB 2
Give a definition for each of the following terms. (i)
Parallel lines
(ii) Perpendicular lines(iii) line segment
(iv) radius
(i)
Parallel lines : Lines which don’t intersect any where are called parallel lines.
(ii)
perpendicular lines : Two lines which are at a right angle to each other are called perpendicular lines,
(iii)
Line segment : It is a terminated line.
(iv)
Radius : The length of the line-segment joining the centre of a circle to any point on its circumference is called it centre.
Theorem 1
: If m,n are lines in the same plane such that intersects m n || m, then intersects n also.
Given : Three lines , m, n in the same plane such that intersect m and n || m.
l
To prove : Lines and n are intersecting lines Proof : Let and n be non intersecting lines. Then || n But,
m
|| m || n and n || m || m and m are non intersecting lines.
This is a contradiction to the hypothesis that and m are intersecting lines.
n
So, our supposition is wrong. Hence, line intersects line n. Theorem 2: If lines AB, AC, AD and AE are parallel to a line , then A, B, C, D and E are collinear. Given : Lines AB, CD, AD and AE are parallel to a line . To prove : A, B, C, D, E are collinear. Proof : Since AB, AC, AD and AE are all parallel to a line . Therefore point A is outside and lines AB, AC, AD, AE are drawn through A and each line is parallel to . But by parallel lines axiom, one and only one line can
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be drawn through A outside it and parallel to . This is possible only when A, B, C, D and E all lie on the same line. Hence, A, B, C, D and E are collinear.
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EXERCISE 1.
2.
(CO-ORDINATE GEOMETRY)
See figure and write the following : (i)
The coordinates of B.
(ii)
The coordinates of C.
(iii)
The point identified by the coordinates (-3, -5)
(iv)
The point identified by the coordinates (2, -4)
(v)
The abscissa of the point D.
(vi)
The ordinate of the point H.
(vii)
The coordinates of the point L.
(viii)
The coordinates of the point M.
See figure and complete the following statements.
(i)
The abscissa and the ordinate of the point B are ____ and ____ respectively. Hence the coordinate of B are (______,______)
(ii)
The x-coordinate and y-coordinate of the point M are ____ and ____ respectively. Hence the coordinate of M are (_____,_______)
(iii)
The x-coordinate and y-coordinate of the point L are ____ and ____ respectively. Hence the coordinate Of L are (______,_______)
(vi)
the x-coordinate and y-coordinate of the point S are ____ and ____ respectively. Hence the coordinate of S are (______,_______)
3.
Write the answer of each of the following questions : (i)
What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane ?
(ii)
What is the name of each part of the plane formed by these two lines ?
(iii)
Write the name of the point where these two lines intersect.
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4.
Plot the point (x, y) given in the following table on the plane, choosing suitable units of distance of the axes. X
-2
-1
0
1
3
Y
8
7
-1.25
3
-1
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5.
Locate the points (5, 0), (0, 5), (2, 5), (5, 2), (-3, 5), (-3, -5), (5, -3) and (6, 1) in the cartesian plane.
6.
Plot the following pairs of number as points in the cartesian plane. Use the scale 1 cm = 1 unit on the axes
7.
x
-3
0
-1
4
2
y
7
-3.5
-3
4
-3
Plot the following points in rectangular coordinate system. In which quadrant do they lie ? (i)
(4, 5)
(v) (-7, 5)
(ii) (4, -5)
(iii) (-10, 2)
(iv) (-10 , -2)
(vi) (9, -3).
8.
Plot the point (-1, 0), (1, 0), (1, 1), (0, 2), (-1, 1) and join them in order. What figure do you get /
9.
Draw the quadrilateral whose vertices are ;
10.
11.
(i)
(1, 1), (2, 4) (8, 4) and (10, 1)
(ii)
(-2, -2), (-4, 2), (-6, -2) and (-4, -6).
In which quadrant will the point lie, if : (i)
The y-coordinate is -3 and the x-coordinate is 4?
(ii)
The x-coordinate is -5 and they y-coordinate is -3 ?
(iii)
The y-coordinate is 4 and the x-coordinate is 5 ?
(iv)
The y-coordinate is 4 and the x-coordinate is -4 ?
Name the quadrant in which the following point lie : (i) P(4, 4)
12.
(ii) Q(-4, 4)
(iii) R(-4, -4)
(iv) S(4, -4)
Plot the point (0, 0), (2, 3), (-2, 3), (-4, -3) and (5, -1) in a rectangular co-ordinate system.
ANSWER KEY
CO-ORDINATE GEOMETRY 1. (i) B (-5, 2), (ii) C (5, -5), 2. (i) 4, 3, (4, 3),
(ii) -3, 4 (-3, 4),
3. (i) The x-axis and the y-axis,
(iii) E, (iv) G, (v) 6,
EXERCISE (IX) CBSE (vi) -3, (vii) L (0, 5) , (viii) M (-3, 0)
(iii) -5, -4, (-5, -4),
(iv) 3, -4(3, -4)
(ii) Quadrants, (iii) The origin.
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10. (i) IV
(ii) III,
(iii) I,
(iv) II
11. (i) I,
(ii) II,
(iii) III,
(iv) IV.
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EXERCISE 1.
2.
(INTRODUCTION TO EUCLID’S GEOMETRY)
Define the following terms : (i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(i) How many lines can pass through a given point ? (ii) In how many points can two distinct lines at the most intersect ?
3.
(i) Given two points P and Q, find how many lines segments do they determine. (ii) Name the lines segments determines by the three collinear points P, Q and R
4.
C
A, B and C are three collinear points such that point A lines between B and C. Name all the line segments determined by these points and write the
B
A
relations between them. 5.
Write the true and false value (T/F) of each of the following statements :(i)
A point is a undefined term.
(ii)
A line is a defined term.
(iii)
Two distinct lines always intersect at one point.
iv)
Two distinct points always determine a line.
(v)
A ray can be extended infinitely on both the sides of it.
(vi)
A line segment has both of its end-points fixed and so it has definite length.
(vii)
Two lines may intersect in two points.
(ix)
A segment has no length.
(x)
Every ray has a finite length.
(xi)
A ray has one end-point only.
(xii)
The ray AB is same as ray BA.
(xiii)
Only a single line may pass through a given point.
(xiv)
Two lines are coincident if they have only one point in common.
6.
Name three undefined terms.
7.
If AB is a line and P is a fixed point, outside AB, how many lines can be drawn through P which are: (i) Parallel to AB (ii) Not parallel to AB.
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8.
Out of the three lines AB, CD and EF, if AB is parallel to EF and CD is also parallel to EF, then what is the relation between AB and CD.
9.
If A, B and C are three points on a line, and B lies between A and C, then prove that :AB + BC = AC.
D C B A 10.
In the given figure, if AB = CD ; prove that: AC = BD.
11.
(i)
How many lines can be drawn to pass through three given points if they are not collinear ?
(ii)
How many line segments can be drawn to pass through two given points if they are collinear ?
12.
Fill in the blanks so as to make the following statements true :(i)
Two distinct points in a place determine a _____ line.
(ii)
Two distinct _____ in a place cannot have more that one point in common.
(iii)
Given a line and a point, not on the line, there is one and only ____ line which passes through the given point and is ____ to the given line.
(iv).
A line separated a plane into ____ pars namely the ____ and the ____ itself.
INTRODUCTION TO EUCLID’S GEOMETRY 2. (i) Infinitely many
(ii) Only one
3. (i) One
(ii) PQ, QR, PR
ANSWER KEY
EXERCISE (IX)- CBSE
4. BA, AC & BC ; BA + AC = BC 5. (i) True
(ii) False
(iii) False
(iv) True
(v) False
(vi) True
(vii)
False
(viii) False
(ix) False
(x) False
(xi) True
(xii) False
(xiii) False
(xiv) False
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6. Point, line and plane
7. (i) Only one, (ii) Infinite
11. (i) Three lines
(ii) one
12. (i) Unique
(ii) Lines
8.
(iii) One, Perpendicular
AB || CD
(iv) Three, two half planes, line.
Important Notes
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