Maths in Focus - Margaret Grove - ch4

4

Geometry 1 TERMINOLOGY Altitude: Height. Any line segment from a vertex to the opposite side of a polygon that is perpendicular to that side

Polygon: General term for a many sided plane figure. A closed plane (two dimensional) figure with straight sides

Congruent triangles: Identical triangles that are the same shape and size. Corresponding sides and angles are equal. The symbol is /

Quadrilateral: A four-sided closed figure such as a square, rectangle, trapezium etc.

Interval: Part of a line including the endpoints

Similar triangles: Triangles that are the same shape but different sizes. The symbol is zy

Median: A line segment that joins a vertex to the opposite side of a triangle that bisects that side

Vertex: The point where three planes meet. The corner of a figure

Perpendicular: A line that is at right angles to another line. The symbol is =

Vertically opposite angles: Angles that are formed opposite each other when two lines intersect

Chapter 4 Geometry 1

INTRODUCTION GEOMETRY IS USED IN many areas, including surveying, building and graphics.

These fields all require a knowledge of angles, parallel lines and so on, and how to measure them. In this chapter, you will study angles, parallel lines, triangles, types of quadrilaterals and general polygons. Many exercises in this chapter on geometry need you to prove something or give reasons for your answers. The solutions to geometry proofs only give one method, but other methods are also acceptable.

DID YOU KNOW? Geometry means measurement of the earth and comes from Greek. Geometry was used in ancient civilisations such as Babylonia. However, it was the Greeks who formalised the study of geometry, in the period between 500 BC and AD 300.

Notation In order to show reasons for exercises, you must know how to name figures correctly. •B The point is called B.

The interval (part of a line) is called AB or BA.

If AB and CD are parallel lines, we write AB < CD.

This angle is named +BAC or +CAB. It can sometimes be named +A. ^

Angles can also be written as BAC or

This triangle is named 3ABC.

BAC

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Maths In Focus Mathematics Preliminary Course

To name a quadrilateral, go around it: for example, BCDA is correct, but ACBD is not.

Producing a line is the same as extending it.

This quadrilateral is called ABCD.

Line AB is produced to C.

+ABD and +DBC are equal.

DB bisects +ABC.

AM is a median of D ABC.

AP is an altitude of D ABC.

Types of Angles Acute angle

0c1 xc1 90c


Right angle

A right angle is 90c. Complementary angles are angles whose sum is 90c.

Obtuse angle

90c1 xc1180c

Straight angle

A straight angle is 180c. Supplementary angles are angles whose sum is 180c.

Reflex angle

180c1 xc1 360c

Angle of revolution

An angle of revolution is 360c.

Vertically opposite angles

+AEC and +DEB are called vertically opposite angles. +AED and +CEB are also vertically opposite angles.

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140


Vertically opposite angles are equal. That is, +AEC = +DEB and +AED = +CEB.

Proof Let +AEC = xc Then +AED = 180c - xc (+CED straight angle, 180c) Now +DEB = 180c - (180c - xc) (+AEB straight angle, 180c) = xc Also +CEB = 180c - xc (+CED straight angle, 180c) ` +AEC = +DEB and +AED =+CEB

EXAMPLES Find the values of all pronumerals, giving reasons. 1.

Solution x + 154 = 180 (+ABC is a straight angle, 180c) x + 154 - 154 = 180 - 154 ` x = 26 2.

Solution 2x + 142 + 90 2x + 232 2x + 232 - 232 2x 2x 2 x

= 360 (angle of revolution, 360c ) = 360 = 360 - 232 = 128 128 = 2 = 64


3.

Solution y + 2y + 30 = 90 3y + 30 3y + 30 - 30 3y 3y 3 y

(right angle, 90c)

= 90 = 90 - 30 = 60 60 3 = 20 =

4.

Solution x + 50 = 165 x + 50 - 50 = 165 - 50 x = 115 y = 180 - 165 = 15 w = 15

(+WZX and +YZV vertically opposite)

(+XZY straight angle, 180c) (+WZY and +XZV vertically opposite)

5.

CONTINUED

141

142


Solution a = 90 b + 53 + 90 = 180 b + 143 = 180 b + 143 - 143 = 180 - 143 b = 37 d = 37 c = 53

(vertically opposite angles) (straight angle, 180c)

(vertically opposite angles) (similarly)

6. Find the supplement of 57c 12l.

Solution Supplementary angles add up to 180c. So the supplement of 57c 12l is 180c - 57c 12l = 122c 48l. 7. Prove that AB and CD are straight lines. A

D

(x + 30)c C

(6x + 10)c

(2x 2 + 10)c E (5x + 30)c B

Solution 6x + 10 + x + 30 + 5x + 30 + 2x + 10 = 360 ^ angle of revolution h 14x + 80 - 80 = 360 - 80 14x = 280 14x 280 = 14 14 x = 20 +AEC = (20 + 30)c = 50c +DEB = (2 # 20 + 10)c = 50c These are equal vertically opposite angles. ` AB and CD are straight lines


4.1 Exercises 1.

Find values of all pronumerals, giving reasons. (a)

yc

(i)

133c

(b)

(j)

(c) 2.

Find the supplement of (a) 59c (b) 107c 31l (c) 45c 12l

3.

Find the complement of (a) 48c (b) 34c 23l (c) 16c 57l

4.

Find the (i) complement and (ii) supplement of (a) 43c (b) 81c (c) 27c (d) 55c (e) 38c (f) 74c 53l (g) 42c 24l (h) 17c 39l (i) 63c 49l (j) 51c 9l

5.

(a) Evaluate x. (b) Find the complement of x. (c) Find the supplement of x.

(d)

(e)

(f)

(g)

(h) (2x + 30)c 142c

143

144


6.

Find the values of all pronumerals, giving reasons for each step of your working.

8.

(a) Prove that CD bisects +AFE. Prove that AC is a straight line.

9.

D C

(b) (3x + 70)c (110 - 3x)c B

(c) A

10. Show that +AED is a right angle. A

(d)

B

(50 - 8y)c

(e)

C

(5y - 20)c

E

(f)

7.

Prove that AC and DE are straight lines.

(3y + 60)c

D


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Parallel Lines When a transversal cuts two lines, it forms pairs of angles. When the two lines are parallel, these pairs of angles have special properties.

Alternate angles

Alternate angles form a Z shape. Can you find another set of alternate angles?

If the lines are parallel, then alternate angles are equal.

Corresponding angles

Corresponding angles form an F shape. There are 4 pairs of corresponding angles. Can you find them?

If the lines are parallel, then corresponding angles are equal.

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Cointerior angles Cointerior angles form a U shape. Can you find another pair?

If the lines are parallel, cointerior angles are supplementary (i.e. their sum is 180c).

Tests for parallel lines

If alternate angles are equal, then the lines are parallel.

If +AEF = +EFD, then AB < CD.

If corresponding angles are equal, then the lines are parallel.

If +BEF = +DFG, then AB < CD.

If cointerior angles are supplementary, then the lines are parallel.

If +BEF + +DFE = 180c, then AB < CD.


If 2 lines are both parallel to a third line, then the 3 lines are parallel to each other. That is, if AB < CD and EF < CD, then AB < EF.

EXAMPLES 1. Find the value of y, giving reasons for each step of your working.

Solution +AGF = 180c - 125c = 55c

(+FGH is a straight angle)

`

(+AGF, +CFE corresponding angles, AB < CD)

y = 55c

2. Prove EF < GH.

Solution +CBF = 180c - 120c (+ABC is a straight angle) = 60c ` +CBF = +HCD = 60c But +CBF and +HCD are corresponding angles ` EF < GH

Can you prove this in a different way?

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148


Think about the reasons for each step of your calculations.

4.2 Exercises 1.

Find values of all pronumerals. (a)

(h)

(i)

(b) (j)

(c)

2.

Prove AB < CD. (a)

(d)

(b) (e)

(c)

A

(f)

(g)

B

104c

C 76c

D

E


A

(d)

(e)

B 138c

B

52c

E C

C

E 128c

D

23c

F 115c

G

H

F

Types of Triangles Names of triangles A scalene triangle has no two sides or angles equal.

A right (or right-angled) triangle contains a right angle.

The side opposite the right angle (the longest side) is called the hypotenuse. An isosceles triangle has two equal sides. The angles (called the base angles) opposite the equal sides in an isosceles triangle are equal.

An equilateral triangle has three equal sides and angles.

A D

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All the angles are acute in an acute-angled triangle.

An obtuse-angled triangle contains an obtuse angle.

Angle sum of a triangle

The sum of the interior angles in any triangle is 180c, that is, a + b + c = 180

Proof

Let +YXZ = ac, +XYZ = bc and +YZX = cc Draw line AB < YZ Then +BXZ = cc (+BXZ, +XZY alternate angles, AB < YZ) +AXY = bc (similarly) +YXZ + +AXY + +BXZ = 180c (+AXB is a straight angle) ` a + b + c = 180


Class Investigation 1. 2. 3. 4. 5.

Could you prove the base angles in an isosceles triangle are equal? Can there be more than one obtuse angle in a triangle? Could you prove that each angle in an equilateral triangle is 60c? Can a right-angled triangle be an obtuse-angled triangle? Can you find an isosceles triangle with a right angle in it?

Exterior angle of a triangle

The exterior angle in any triangle is equal to the sum of the two opposite interior angles. That is, x+y=z

Proof

Let +ABC = xc , +BAC = yc and +ACD = zc Draw line CE < AB zc = +ACE + +ECD +ECD = xc +ACE = yc ` z=x+y

(+ECD,+ABC corresponding angles, AB < CE) (+ACE,+BAC alternate angles, AB < CE)

EXAMPLES Find the values of all pronumerals, giving reasons for each step. 1.

CONTINUED

151

152


Solution x + 53 + 82 = 180 (angle sum of D 180c) x + 135 = 180 x + 135 - 135 = 180 - 135 x = 45 2.

Solution +A = +C = x x + x + 48 = 180 2x + 48 = 180 2x + 48 - 48 = 180 - 48 2x = 132 132 2x = 2 2 x = 66

(base angles of isosceles D) (angle sum in a D 180c)

3.

Solution y + 35 = 141 (exterior angle of D) y + 35 - 35 = 141 - 35 ` y = 106 This example can be done using the interior sum of angles. +BCA = 180c - 141c = 39c y + 39 + 35 = 180 y + 74 = 180 y + 74 - 74 = 180 - 74 ` y = 106

(+BCD is a straight angle 180c) (angle sum of D 180c)


Think of the reasons for each step of your calculations.

4.3 Exercises 1.

Find the values of all pronumerals. (a)

(h)

(b) (i)

(j) (c)

(d)

(k)

(e)

(f)

(g)

153

2.

Show that each angle in an equilateral triangle is 60c.

3.

Find +ACB in terms of x.

154


4.

5.

6.

Prove AB < ED.

(d)

8.

Prove D IJL is equilateral and D JKL is isosceles.

9.

In triangle BCD below, BC = BD. Prove AB ED.

Show D ABC is isosceles.

Line CE bisects +BCD. Find the value of y, giving reasons.

A B C

46c E 88c

D

7.

Evaluate all pronumerals, giving reasons for your working. (a)

10. Prove that MN QP . 32c

M

(b) 75c

O

73c

Q

(c)

P

N


Congruent Triangles Two triangles are congruent if they are the same shape and size. All pairs of corresponding sides and angles are equal. For example:

We write D ABC / D XYZ.

Tests To prove that two triangles are congruent, we only need to prove that certain combinations of sides or angles are equal.

Two triangles are congruent if • SSS: all three pairs of corresponding sides are equal • SAS: two pairs of corresponding sides and their included angles are equal • AAS: two pairs of angles and one pair of corresponding sides are equal • RHS: both have a right angle, their hypotenuses are equal and one other pair of corresponding sides are equal

EXAMPLES 1. Prove that DOTS / DOQP where O is the centre of the circle.

CONTINUED

The included angle is the angle between the 2 sides.

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156


Solution S: A: S:

OS = OQ +TOS = +QOP OT = OP

`

by SAS, DOTS / DOQP

(equal radii) (vertically opposite angles) (equal radii)

2. Which two triangles are congruent?

Solution To find corresponding sides, look at each side in relation to the angles. For example, one set of corresponding sides is AB, DF, GH and JL. D ABC / D JKL (by SAS) 3. Show that triangles ABC and DEC are congruent. Hence prove that AB = ED.

Solution A: +BAC = +CDE A: +ABC = +CED S: AC = CD

(alternate angles, AB < ED) (similarly) (given)

` by AAS, D ABC / D DEC ` AB = ED

(corresponding sides in congruent D s)


4.4 Exercises 1.

Are these triangles congruent? If they are, prove that they are congruent. (a)

2.

Prove that these triangles are congruent. (a)

B

(b)

Y 4.7

m

110c

2.3

4.7

m

m

Z

110 c C

A 2

.3 m

(b)

X

(c)

(c)

(d)

(d) (e)

(e)

157

158


3.

A

Prove that (a) Δ ABD is congruent to Δ ACD (b) AB bisects BC, given D ABC is isosceles with AB = AC.

D

B

4.

Prove that triangles ABD and CDB are congruent. Hence prove that AD = BC.

C

(a) Prove that TABC and TADC are congruent. (b) Show that +ABC = +ADC. The centre of a circle is O and AC is perpendicular to OB.

7.

A

5.

In the circle below, O is the centre of the circle. A

O

D

B O

C

B

C

(a) Prove that TOAB and TOCD are congruent. (b) Show that AB = CD. 6.

(a) Show that TOAB and TOBC are congruent. (b) Prove that +ABC = 90c. ABCF is a trapezium with AF = BC and FE = CD. AE and BD are perpendicular to FC.

8.

In the kite ABCD, AB = AD and BC = DC.

F

A

B

E

D

C

(a) Show that TAFE and TBCD are congruent. (b) Prove that +AFE = +BCD.


9.

The circle below has centre O and OB bisects chord AC.

10. ABCD is a rectangle as shown below. A

B

D

C

C O B

A

(a) Prove that TOAB is congruent to TOBC. (b) Prove that OB is perpendicular to AC.

(a) Prove that TADC is congruent to TBCD. (b) Show that diagonals AC and BD are equal.

Investigation The triangle is used in many structures, for example trestle tables, stepladders and roofs. Find out how many different ways the triangle is used in the building industry. Visit a building site, or interview a carpenter. Write a report on what you find.

Similar Triangles Triangles, for example ABC and XYZ, are similar if they are the same shape but different sizes. As in the example, all three pairs of corresponding angles are equal. All three pairs of corresponding sides are in proportion (in the same ratio).

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We write: D ABC <; D XYZ D XYZ is three times larger than D ABC. 6 XY = =3 AB 2 XZ 12 = =3 4 AC 15 YZ = =3 5 BC XY XZ YZ ` = = AB AC BC

This shows that all 3 pairs of sides are in proportion.

Application Similar figures are used in many areas, including maps, scale drawings, models and enlargements.

EXAMPLE 1. Find the values of x and y in similar triangles CBA and XYZ.

Solution First check which sides correspond to one another (by looking at their relationships to the angles). YZ and BA, XZ and CA, and XY and CB are corresponding sides. `

XZ XY = CA CB y 5.4 = 4.9 3.6 3.6y = 4.9 # 5.4


161

4 . 9 # 5 .4 3 .6 = 7.35 XY = CB 5 .4 = 3.6 = 2 . 3 # 5 .4 2 . 3 # 5 .4 = 3 .6 = 3.45

y= YZ BA x 2 .3 3 .6x x

Tests There are three tests for similar triangles. Two triangles are similar if: • three pairs of corresponding angles are equal • three pairs of corresponding sides are in proportion • two pairs of sides are in proportion and their included angles are equal

EXAMPLES 1. (a) Prove that triangles ABC and ADE are similar. (b) Hence find the value of y, to 1 decimal place.

Solution (a) +A is common +ABC = +ADE +ACB = +AED ` D ABC <; D ADE (b)

(corresponding angles, BC < DE) (similarly) (3 pairs of angles equal)

CONTINUED

If 2 pairs of angles are equal then the third pair must also be equal.

162


AE = 2.4 + 1.9 = 4 .3 DE AE = BC AC y 4 .3 = 3 .7 2.4 2 .4 y = 3 . 7 # 4. 3 3 .7 # 4 .3 y= 2 .4 = 6 .6 2. Prove D XYZ <; DWVZ.

Solution XZ ZV YZ ZW XZ ` ZV +XZY

3 15 = 7 35 3 6 = = 7 14 YZ = ZW = +WZV =

(vertically opposite angles)

` since two pairs of sides are in proportion and their included angles are equal the triangles are similar

Ratio of intercepts The following result comes from similar triangles.

When two (or more) transversals cut a series of parallel lines, the ratios of their intercepts are equal. That is, AB : BC = DE : EF AB DE or = EF BC


Proof Draw DG and EH parallel to AC.

Then Also `

`

DG = AB EH = BC DG AB = EH BC +GDE = +HEF +DEG = +EFH +DGE = +EHF

(opposite sides of a parallelogram) (similarly) (1) (corresponding +s, DG < EH) (corresponding +s, BE < CF) (angle sum of Ds)

So D DGE <; D EHF DG DE = ` EH EF From (1) and (2):

(2)

AB DE = EF BC

EXAMPLES 1. Find the value of x, to 3 significant figures.

Solution x 1.5 = ^ ratios of intercepts on parallel lines h 8.9 9.3 9.3x = 8.9 # 1.5 8.9 # 1.5 x= 9.3 = 1.44

CONTINUED

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2. Evaluate x and y, to 1 decimal place.

Solution Use either similar triangles or ratios of intercepts to find x. You must use similar triangles to find y.

Why?

x 2.7 = 5.8 3. 4 2.7 # 5.8 x= 3.4 = 4.6 y 2.7 + 3.4 = 7.1 3.4 6.1 # 7.1 y= 3.4 = 12.7

These ratios come from intercepts on parallel lines.

These ratios come from similar triangles.

4.5 Exercises 1.

Find the value of all pronumerals, to 1 decimal place where appropriate. (a)

(c)

(d)

(e) (b)


(f) 46 c

xc

11

8.9 25.7

9.1

1.3

5c

c

1.82 E

14.3

19

c

4.

52c

A

4.2

B

4.9 5.88

yc

6.86

F

The diagram shows two concentric circles with centre O. (a) Prove that DOAB <; D OCD. (b) If radius OC = 5.9 cm and radius OB = 8.3 cm, and the length of CD = 3.7 cm, find the length of AB, correct to 2 decimal places.

7.

(a) Prove that D ABC <; D ADE. (b) Find the values of x and y, correct to 2 decimal places.

8.

ABCD is a parallelogram, with CD produced to E. Prove that D ABF <; DCEB.

Evaluate a and b to 2 decimal places.

Show that D ABC and DCDE are similar.

EF bisects +GFD. Show that D DEF and D FGE are similar.

C

6.

(g)

3.

D 87c

y

46

a

2.

Show that D ABC and D DEF are similar. Hence find the value of y.

5.

165

166


9.

Show that D AED <; D ABC. Find the value of m.

10. Prove that D ABC and D ACD are similar. Hence evaluate x and y.

(e)

12. Show that AB AF (a) = BC FG AB AF (b) = AC AG BD DF (c) = CE EG

11. Find the values of all pronumerals, to 1 decimal place. (a) 13. Evaluate a and b correct to 1 decimal place.

(b)

14. Find the value of y to 2 significant figures. (c)

(d) 15. Evaluate x and y correct to 2 decimal places.


Pythagoras’ Theorem DID YOU KNOW? The triangle with sides in the proportion 3:4:5 was known to be right angled as far back as ancient Egyptian times. Egyptian surveyors used to measure right angles by stretching out a rope with knots tied in it at regular intervals. They used the rope for forming right angles while building and dividing fields into rectangular plots. It was Pythagoras (572–495 BC) who actually discovered the relationship between the sides of the right-angled triangle. He was able to generalise the rule to all right-angled triangles. Pythagoras was a Greek mathematician, philosopher and mystic. He founded the Pythagorean School, where mathematics, science and philosophy were studied. The school developed a brotherhood and performed secret rituals. He and his followers believed that the whole universe was based on numbers. Pythagoras was murdered when he was 77, and the brotherhood was disbanded.

The square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other two sides. That is, c2 = a2 + b2 or

c=

a2 + b2

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168


Proof

Draw CD perpendicular to AB Let AD = x, DB = y Then x + y = c In D ADC and D ABC, +A is common +ADC = +ACB = 90c `

D ADC <; D ABC (equal corresponding +s) AC AD = AB AC x b c =b b 2 = xc D BDC <; D ABC Similarly, BC DB = AB BC y a a= c a 2 = yc Now a 2 + b 2 = yc + xc = c ^y + xh = c ]c g = c2 If c 2 = a 2 + b 2, then D ABC must be right angled

EXAMPLES 1. Find the value of x, correct to 2 decimal places.

Solution c2 = a2 + b2 x2 = 72 + 42 = 49 + 16 = 65


169

x = 65 = 8.06 to 2 decimal places 2. Find the exact value of y.

Solution c2 = a2 + b2 82 = y2 + 42 64 = y 2 + 16 48 = y 2 `

Leave the answer in surd form for the exact answer.

y = 48 = 16 # 3 =4 3

3. Find the length of the diagonal in a square with sides 6 cm. Answer to 1 decimal place.

Solution

6 cm

6 cm

c =a +b = 62 + 62 = 72 2

2

2

c = 72 = 8 .5 So the length of the diagonal is 8.5 cm.

CONTINUED

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4. A triangle has sides 5.1 cm, 6.8 cm and 8.5 cm. Prove that the triangle is right angled.

Solution

5.1 cm

8.5 cm

6.8 cm

Let c = 8.5 (largest side) and a and b the other two smaller sides. a 2 + b 2 = 5 . 1 2 + 6. 8 2 = 72.25 c 2 = 8. 5 2 = 72.25 ` c2 = a2 + b2 So the triangle is right angled.

4.6 Exercises 1.

Find the value of all pronumerals, correct to 1 decimal place. (a)

2.

Find the exact value of all pronumerals. (a)

(b) (b)

(c) (c)

(d) (d)


3.

Find the slant height s of a cone with diameter 6.8 m and perpendicular height 5.2 m, to 1 decimal place.

4.

Find the length of CE, correct to 1 decimal place, in this rectangular pyramid. AB = 8.6 cm and CF = 15.9 cm.

5.

Prove that D ABC is a right-angled triangle.

6.

7.

Show that AC = 2 BC.

8.

(a) Find the length of diagonal AC in the figure. (b) Hence, or otherwise, prove that AC is perpendicular to DC.

9.

Find the length of side AB in terms of b.

Show that D XYZ is a right-angled isosceles triangle. X

XY in YZ terms of x and y in D XYZ.

10. Find the exact ratio of 2

1

Y

1

Z

171

172


11. Show that the distance squared between A and B is given by d 2 = 13t 2 - 180t + 625.

16. A ramp is 4.5 m long and 1.3 m high. How far along the ground does the ramp go? Answer correct to one decimal place. 12. An 850 mm by 1200 mm gate is to have a diagonal timber brace to give it strength. To what length should the timber be cut, to the nearest mm?

4.5 m

1.3 m

17. The diagonal of a television screen is 72 cm. If the screen is 58 cm high, how wide is it? 18. A property has one side 1.3 km and another 1.1 km as shown with a straight road diagonally through the middle of the property. If the road is 1.5 km long, show that the property is not rectangular.

13. A rectangular park has a length of 620 m and a width of 287 m. If I walk diagonally across the park, how far do I walk? 14. The triangular garden bed below is to have a border around it. How many metres of border are needed, to 1 decimal place?

1.5 km

1.3 km

1.1 km

15. What is the longest length of stick that will fit into the box below, to 1 decimal place?

19. Jodie buys a ladder 2 m long and wants to take it home in the boot of her car. If the boot is 1.2 m by 0.7 m, will the ladder fit?


20. A chord AB in a circle with centre O and radius 6 cm has a perpendicular line OC as shown 4 cm long.

(a) By finding the lengths of AC and BC, show that OC bisects the chord. (b) By proving congruent triangles, show that OC bisects the chord.

O A

4 cm

6 cm

C B

Types of Quadrilaterals A quadrilateral is any four-sided figure

In any quadrilateral the sum of the interior angles is 360c

Proof Draw in diagonal AC +ADC + +DCA + +CAD = 180c (angle sum of D) +ABC + +BCA + +CAB = 180c (similarly) ` +ADC + +DCA + +CAD + +ABC + +BCA + +CAB = 360c That is, +ADC + +DCB + +CBA + +BAD = 360c

173

174


EXAMPLE Find the value of i.

Solution i + 120 + 56 + 90 = 360 ^ angle sum of quadrilateral h i + 266 = 360 i = 94

Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel PROPERTIES

• • • •

These properties can all be proven.

opposite sides of a parallelogram are equal opposite angles of a parallelogram are equal diagonals in a parallelogram bisect each other each diagonal bisects the parallelogram into two congruent triangles

TESTS A quadrilateral is a parallelogram if: • both pairs of opposite sides are equal • both pairs of opposite angles are equal • one pair of sides is both equal and parallel • the diagonals bisect each other


175

Rectangle

A rectangle is a parallelogram with one angle a right angle

If one angle is a right angle, then you can prove all angles are right angles.

PROPERTIES • the same as for a parallelogram, and also • diagonals are equal

TEST A quadrilateral is a rectangle if its diagonals are equal

Application Builders use the property of equal diagonals to check if a rectangle is accurate. For example, a timber frame may look rectangular, but may be slightly slanting. Checking the diagonals makes sure that a building does not end up like the Leaning Tower of Pisa!

Rhombus

A rhombus is a parallelogram with a pair of adjacent sides equal PROPERTIES • the same as for parallelogram, and also • diagonals bisect at right angles • diagonals bisect the angles of the rhombus

It can be proved that all sides are equal.

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TESTS

A quadrilateral is a rhombus if: • all sides are equal • diagonals bisect each other at right angles

Square

A square is a rectangle with a pair of adjacent sides equal PROPERTIES

• the same as for rectangle, and also • diagonals are perpendicular • diagonals make angles of 45c with the sides

Trapezium

A trapezium is a quadrilateral with one pair of sides parallel

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal


EXAMPLES 1. Find the values of i, x and y, giving reasons.

Solution i = 83c x = 6.7 cm y = 2.3 cm

(opposite +s in < gram) (opposite sides in < gram) (opposite sides in < gram)

2. Find the length of AB in square ABCD as a surd in its simplest form if BD = 6 cm.

Solution Let AB = x Since ABCD is a square, AB = AD = x (adjacent sides equal) Also, +A = 90c (by definition) By Pythagoras’ theorem: c2 = a2 + b2 62 = x2 + x2 36 = 2x 2 18 = x 2 ` x = 18 = 3 2 cm

CONTINUED

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3. (a) Two equal circles have centres O and P respectively. Prove that OAPB is a rhombus. (b) Hence, or otherwise, show that AB is the perpendicular bisector of OP.

Solution OA = OB (equal radii) PA = PB (similarly) Since the circles are equal, OA = OB = PA = PB ` since all sides are equal, OAPB is a rhombus (b) The diagonals in any rhombus are perpendicular bisectors. Since OAPB is a rhombus, with diagonals AB and OP, AB is the perpendicular bisector of OP.

(a)

4.7 Exercises 1.

Find the value of all pronumerals, giving reasons. (a)

(e)

(f) (b)

(g) (c)

(d)


2.

Given AB = AE, prove CD is perpendicular to AD.

(c)

(d)

3.

(a) Show that +C = xc and +B = +D = (180 - x)c. (b) Hence show that the sum of angles of ABCD is 360c.

(e)

(f) 4.

5.

Find the value of a and b.

7

3x

x+

6

y

6.

In the figure, BD bisects +ADC. Prove BD also bisects +ABC.

7.

(a)

Prove that each figure is a parallelogram. (a)

(b)

(b)

Find the values of all pronumerals, giving reasons.

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180


(c)

(d)

(e) (d)

8.

Evaluate all pronumerals. (a)

9.

The diagonals of a rhombus are 8 cm and 10 cm long. Find the length of the sides of the rhombus.

10. ABCD is a rectangle with +EBC = 59c . Find +ECB, +EDC and +ADE. (b)

(c)

11. The diagonals of a square are 8 cm long. Find the exact length of the side of the square. 12. In the rhombus, +ECB = 33c. Find the value of x and y.

ABCD is a kite

Polygons A polygon is a closed plane figure with straight sides

A regular polygon has all sides and all interior angles equal


EXAMPLES

3-sided (equilateral triangle)

4-sided (square)

5-sided (pentagon)

6-sided (hexagon)

8-sided (octagon)

10-sided (decagon)

DID YOU KNOW? Carl Gauss (1777–1855) was a famous German mathematician, physicist and astronomer. When he was 19 years old, he showed that a 17-sided polygon could be constructed using a ruler and compasses. This was a major achievement in geometry. Gauss made a huge contribution to the study of mathematics and science, including correctly calculating where the magnetic south pole is and designing a lens to correct astigmatism. He was the director of the Göttingen Observatory for 40 years. It is said that he did not become a professor of mathematics because he did not like teaching.

The sum of the interior angles of an n-sided polygon is given by S = 180n - 360 or S = (n - 2) # 180c

Proof Draw any n-sided polygon and divide it into n triangles as shown. Then the total sum of angles is n # 180c or 180n. But this sum includes all the angles at O. So the sum of interior angles is 180n - 360c . That is, S = 180n - 360 = ] n - 2 g #180c

The sum of the exterior angles of any polygon is 360c

Proof Draw any n-sided polygon. Then the sum of both the exterior and interior angles is n #180c. Sum of exterior angles = n #180c - sum of interior angles = 180n - ] 180n - 360c g = 180n - 180n + 360c = 360c

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EXAMPLES 1. Find the sum of the interior angles of a regular polygon with 15 sides. How large is each angle?

Solution n = 15 S = (n - 2)#180c = (15 - 2)#180c = 13#180c = 2340c Each angle has size 2340c' 15 = 156c. 2. Find the number of sides in a regular polygon whose interior angles are 140c.

Solution Let n be the number of sides Then the sum of interior angles is 140n But S = (n - 2)#180c So 140n = (n - 2)#180c = 180n - 360 360 = 40n 9=n So the polygon has 9 sides.

There are n sides and so n angles, each 140c.

4.8 Exercises 1.

2.

Find the sum of the interior angles of (a) a pentagon (b) a hexagon (c) an octagon (d) a decagon (e) a 12-sided polygon (f) an 18-sided polygon Find the size of each interior angle of a regular (a) pentagon (b) octagon (c) 12-sided polygon (d) 20-sided polygon (e) 15-sided polygon

3.

Find the size of each exterior angle of a regular (a) hexagon (b) decagon (c) octagon (d) 15-sided polygon

4.

Calculate the size of each interior angle in a regular 7-sided polygon, to the nearest minute.

5.

The sum of the interior angles of a regular polygon is 1980c. (a) How many sides has the polygon? (b) Find the size of each interior angle, to the nearest minute.


6.

Find the number of sides of a regular polygon whose interior angles are 157c 30l.

7.

Find the sum of the interior angles of a regular polygon whose exterior angles are 18c.

8.

9.

A regular polygon has interior angles of 156c. Find the sum of its interior angles.

13. A regular octagon has a quadrilateral ACEG inscribed as shown. B

D

H

Find the size of each interior angle in a regular polygon if the sum of the interior angles is 5220c.

10. Show that there is no regular polygon with interior angles of 145c. 11. Find the number of sides of a regular polygon with exterior angles (a) 40c (b) 30c (c) 45c (d) 36c (e) 12c

C

A

G

E

F

Show that ACEG is a square. 14. In the regular pentagon below, show that EAC is an isosceles triangle. A

E

B

12. ABCDEF is a regular hexagon. A

B D

F

C

E

D

(a) Show that triangles AFE and BCD are congruent. (b) Show that AE and BD are parallel.

C

15. (a) Find the size of each exterior angle in a regular polygon with side p. (b) Hence show that each interior 180 (p - 2) . angle is p

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Areas Most areas of plane figures come from the area of a rectangle.

Rectangle

A = lb

Square

A square is a special rectangle.

A = x2

Triangle

The area of a triangle is half the area of a rectangle.

A=

1 bh 2

Proof

h

b

Draw rectangle ABCD, where b = length and h = breadth.


185

area = bh

`

1 1 area AEFD and area DCEF = area EBCF 2 2 1 area DCDE = area ABCD 2 1 That is, A = bh 2

Area D DEF = `

Parallelogram

A = bh

Proof In parallelogram ABCD, produce DC to E and draw BE perpendicular to CE. Then ABEF is a rectangle.

Area ABEF = bh In D ADF and D BCE, +AFD = +BEC = 90c AF = BE = h (opposite sides of a rectangle) AD = BC (opposite sides of a parallelogram) ` by RHS, D ADF / D BCE ` area D ADF = area D BCE So area ABCD = area ABEF = bh

Rhombus

1 xy 2 (x and y are lengths of diagonals) A=

The area of a parallelogram is the same as the area of two triangles.

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Proof

Let AC = x and BD = y By properties of a rhombus, 1 1 AE = EC = x and DE = EB = y 2 2 Also +AEB = 90c Area D ABC = = Area D ADC = = ` total area of rhombus = =

1 1 x: y 2 2 1 xy 4 1 1 x: y 2 2 1 xy 4 1 1 xy + xy 4 4 1 xy 2

Trapezium

A=

Proof

Let DE = x Then DF = x + a ` FC = b - ] x + a g =b-x-a

1 h ( a + b) 2


Area trapezium = area D ADE + area rectangle ABFE + area D BFC 1 1 = xh + ah + (b - x - a) h 2 2 1 = h ( x + 2 a + b - x - a) 2 1 = h (a + b ) 2

Circle

A = rr 2

EXAMPLES 1. Find the area of this trapezium.

Solution 1 h ( a + b) 2 1 = ( 4) ( 7 + 5) 2 = 2 # 12

A=

= 24 m 2

4.2 cm

8.9 cm

3.7 cm

2. Find the area of the shaded region in this figure.

12.1 cm CONTINUED

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Solution Area large rectangle = lb = 8.9 #12.1 = 107.69 cm 2 Area small rectangle = lb = 3 . 7 # 4 .2 `

= 15.54 cm 2 shaded area = 107.69 - 15.54 = 92.15 cm 2

3. A park with straight sides of length 126 m and width 54 m has semicircular ends as shown. Find its area, correct to 2 decimal places. 126 m

54 m

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Solution Area of 2 semi-circles = area of 1 circle 54 r = 2 = 27 A = rr 2 = r (27) 2 = 2290.22 m2 Area rectangle = 126 # 54 = 6804 Total area = 2290.22 + 6804 = 9094.22 m2

4.9 Exercises 1.

Find the area of each figure. (a)

(b)


(c)

(b)

(c)

(d)

(e) (d)

(e)

(f)

6

cm

2 cm

(g) 4.

2.

Find the area of a rhombus with diagonals 2.3 m and 4.2 m.

3.

Find each shaded area. (a)

Find the area of each figure. (a)

(b)

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(c)

(d)

8.

The dimensions of a battleaxe block of land are shown below. (a) Find its area. (b) A house in the district where this land is can only take up 55% of the land. How large (to the nearest m 2) can the area of the house be? (c) If the house is to be a rectangular shape with width 8.5 m, what will its length be?

9.

A rhombus has one diagonal 25 cm long and its area is 600 cm 2 . Find the length of (a) its other diagonal and (b) its side, to the nearest cm.

(e)

5.

Find the exact area of the figure.

6.

Find the area of this figure, correct to 4 significant figures. The arch is a semicircle.

7.

Jenny buys tiles for the floor of her bathroom (shown top next column) at $45.50 per m 2 . How much do they cost altogether?

10. The width w of a rectangle is a quarter the size of its length. If the width is increased by 3 units while the length remains constant, find the amount of increase in its area in terms of w.


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Test Yourself 4 1.

Find the values of all pronumerals (a)

2.

Prove that AB and CD are parallel lines.

3.

Find the area of the figure, to 2 decimal places.

4.

(a) Prove that triangles ABC and ADE are similar. (b) Evaluate x and y to 1 decimal place.

5.

Find the size of each interior angle in a regular 20-sided polygon.

6.

Find the volume of a cylinder with radius 5.7 cm and height 10 cm, correct to 1 decimal place.

7.

Find the perimeter of the triangle below.

(b)

(c)

(d)

x

(e) (O is the centre of the circle.)

(f)

(g)

The perimeter is the distance around the outside of the figure.

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8.

(a) Prove triangles ABC and ADC are congruent in the kite below.

(b) Prove triangle AOB and COD are congruent. (O is the centre of the circle.)

12. Triangle ABC is isosceles, and AD bisects BC.

(a) Prove triangles ABD and ACD are congruent. (b) Prove AD and BC are perpendicular. 13. Triangle ABC is isosceles, with AB = AC. Show that triangle ACD is isosceles.

9.

Find the area of the figure below.

14. Prove that opposite sides in any parallelogram are equal.

10. Prove triangle ABC is right angled.

15. A rhombus has diagonals 6 cm and 8 cm. (a) Find the area of the rhombus. (b) Find the length of its side. 16. The interior angles in a regular polygon are 140c . How many sides has the polygon? 17. Prove AB and CD are parallel.

11. Prove

AF AB . = AG AC


18. Find the area of the figure below. 6 cm 5 cm 8 cm

20. (a) Prove triangles ABC and DEF are similar. (b) Evaluate x to 1 decimal place.

2 cm

10 cm

19. Prove that z = x + y in the triangle below.

Challenge Exercise 4 1.

Find the value of x.

4.

Given +BAD =+DBC, show that D ABD and D BCD are similar and hence find d.

2.

Evaluate x, y and z.

5.

Prove that ABCD is a parallelogram. AB = DC.

3.

Find the sum of the interior angles of a regular 11-sided polygon. How large is each exterior angle?

6.

Find the shaded area.

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7.

Prove that the diagonals in a square make angles of 45c with the sides.

8.

Prove that the diagonals in a kite are perpendicular.

9.

Prove that MN is parallel to XY.

12. Find the values of x and y correct to 1 decimal place.

13. Find the values of x and y, correct to 2 decimal places.

10. Evaluate x.

11. The letter Z is painted on a billboard.

(a) Find the area of the letter. (b) Find the exact perimeter of the letter.

14. ABCD is a square and BD is produced to 1 E such that DE = BD. 2 (a) Show that ABCE is a kite. 2x (b) Prove that DE = units when 2 sides of the square are x units long.

Maths in Focus - Margaret Grove - ch4

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