Chapter 13 Geometry 13_NCM10A_94662

Measurement and geometry

13

Geometry The word ‘geometry’ comes from the Greek word geometria which means ‘land measuring’. The principles and ideas of geometry are evident in many aspects of our lives. For example, geometry can be seen in the design of buildings, bridges, roads and transport networks.

N E W C E N T U R Y M AT H S A D V A N C E D ustralian Curriculum

10 þ10A

Shutterstock.com/Sergey Kelin

for the A

n Chapter outline 13-01 Angle sum of a polygon 13-02 Congruent triangle proofs 13-03 Tests for quadrilaterals* 13-04 Proving properties of triangles and quadrilaterals 13-05 Formal geometrical proofs* 13-06 Similar figures 13-07 Finding unknown sides in similar figures 13-08 Tests for similar triangles 13-09 Similar triangle proofs* *STAGE 5.3

9780170194662

n Wordbank Proficiency strands U F R C U U

F F

PS PS

R R

C C

U

F

PS

R

C

U U

F F

PS

R R

C C

U

F

R

C

U U

F F

R R

C C

PS PS

congruence test One of four tests for proving that triangles are congruent: SSS, SAS, AAS and RHS congruent Identical; exactly the same (symbol: ”) included angle The angle between two given sides of a shape quadrilateral test A property of a quadrilateral that proves that it is a particular type of quadrilateral, for example, if its opposite angles are equal, then it must be a parallelogram rectangle A parallelogram with a right angle regular polygon A polygon with all angles equal and all sides equal, such as an equilateral triangle or a square similar To have the same shape but not necessarily the same size, an enlargement or reduction (symbol: |||) similarity test One of four tests for proving that triangles are similar: ‘SSS’, ‘SAS’, ‘AA’ and ‘RHS’

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

n In this chapter you will: • formulate proofs involving congruent triangles and angle properties • solve problems using ratio and scale factors in similar figures • (STAGE 5.3) apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes • solve problems involving the angle sum of a polygon and the exterior angle sum of a convex polygon • write formal proofs for congruent triangles • (STAGE 5.3) understand and apply the definitions and tests for the special quadrilaterals • prove properties of triangles and quadrilaterals using congruent triangles • explain similarity and investigate the properties of similar figures • identify and use the four tests for similar triangles • (STAGE 5.3) write formal proofs of the similarity of triangles

SkillCheck Worksheet StartUp assignment 11

1

Find the value of each pronumeral. a

MAT10MGWK10077

b

c x°

68°

h° 37°

p°

27°

38°

Puzzle sheet Finding angles MAT10MGPS00026

d

e

f 3a°

Video tutorial Geometry

124°

m° 60°

MAT10MGVT00008

g

h

i

k°

Starting Geometer’s SketchPad

82°

76°

r°

4p°

r°

w°

MAT10MGSS10013

3w°

w°

2a°

24 °

Skillsheet

w°

73°

j

k

l

128° 48°

p° y°

488

63°

d°

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

2

ustralian Curriculum

a Find the value of m, n and p, giving reasons. b What type of triangle is n XYW?

10 þ10A

X 115° 65° m°

p°

n°

W

Y

13-01 Angle sum of a polygon

NSW

Worksheet Angles in polygons MAT10MGWK10078

Technology GeoGebra: Naming polygons

Alamy/Raymond Warren

MAT10MGTC00008

A polygon is any shape with straight sides. A polygon may be either convex or non-convex (concave).

Convex polygon

Non-convex polygon

In a convex polygon, all vertices point outwards, all diagonals lie within the shape and all angles are less than 180. In a non-convex polygon, some vertices point inwards, some diagonals lie outside the shape and some angles are more than 180 (reflex angles).

9780170194662

489

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Summary The angle sum of a polygon with n sides is given by the formula A ¼ 180(n  2). This formula applies to both convex and non-convex polygons.

Example

1

Find the angle sum of a 15-sided polygon.

Solution Angle sum ¼ 180ð15  2Þ

n ¼ 15



¼ ð180 3 13Þ ¼ 2340

Example

2

Find the number of sides in a polygon that has an angle sum of 1080.

Solution 180ðn  2Þ ¼ 1080 180n  360 ¼ 1080 180n ¼ 1440 1440 180 ¼8

n¼

[ The polygon has 8 sides (octagon). A regular polygon has all its angles and sides equal. For example, a regular hexagon has 6 equal angles and 6 equal sides. A square is a regular polygon but a rhombus is not.

Summary The size of each angle in a regular polygon with n sides ¼

490

180ðn  2Þ Angle sum ¼ n Number of sides



9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

Example

ustralian Curriculum

10 þ10A

3

Find the size of one angle in a regular pentagon.

Solution A pentagon has 5 sides (n ¼ 5). 180ð5  2Þ 5 ð180 3 3Þ ¼ 5 ¼ 108

Size of one angle ¼

Each angle in a regular pentagon is 108.

Exterior angle sum of a convex polygon Summary The sum of the exterior angles of a convex polygon is 360.

C D B

A

Example

E

4

For a regular octagon, find the size of: a each exterior angle

b each (interior) angle.

Solution a Sum of exterior angles ¼ 360 One exterior angle ¼ 360 4 8 ¼ 45 b Each angle ¼ 180  45 ¼ 135 OR :

9780170194662

ðangles on a straight lineÞ

180ð8  2Þ 8  ¼ 135

Each angle ¼

491

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Example

5

Find the number of sides in a regular polygon if: a each exterior angle is 24

b each (interior) angle is 140.

Solution a Number of exterior angles ¼ 360 4 24 ¼ 15 [ The regular polygon has 15 sides. b Exterior angle ¼ 180  140 ðangles on a straight lineÞ ¼ 40 Sum of exterior angles ¼ 360 Number of exterior angles ¼ 360 4 40 ¼9 [ The regular polygon has 9 sides. OR:

180ðn  2Þ ¼ 140 n 180ðn  2Þ ¼ 140n 180n  360 ¼ 140n 40n  360 ¼ 0 40n ¼ 360

360 40 ¼9 [ The regular polygon has 9 sides. n¼

Exercise 13-01 Angle sum of a polygon See Example 1

See Example 2

See Example 3

1 2

Find the angle sum of a polygon with: a 12 sides b 10 sides c 9 sides

Find the number of sides in a polygon that has an angle sum of: a 720 b 3420 c 1980 d 5040

3

The angle sum of a regular polygon is 2520. a How many sides does the polygon have? b Find the size of each angle.

4

Find the size of one angle in a regular: a decagon

5

b octagon

c hexagon

e 15 sides e 1260

d dodecagon

How many sides does a regular polygon have if each of its angles is: a 168?

492

d 20 sides

b 156?

c 172?

d 165.6?

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

6

8 9

10 þ10A

Find the size of each exterior angle of a regular: a pentagon

7

ustralian Curriculum

b dodecagon

Find the size of each angle in a regular: a nonagon b 20-sided polygon

See Example 4

c 18-sided polygon

d hexagon

c decagon

d 30-sided polygon

Find the number of sides in a regular polygon if each exterior angle is: a 15 b 72 c 20 d 40 e 5

f 12

Find the number of sides in a regular polygon if each angle is: a 135 b 144 c 156 d 178 e 165

f 150

Just for the record

See Example 5

The geometry of Canberra

Canberra is located 300 km south-west of Sydney and was designed by the American architect Walter Burley Griffin. Construction of Australia’s capital city began in 1913. The ‘centre’ of Canberra is based on an equilateral triangle, bounded by the ‘sides’ Commonwealth Avenue, Kings Avenue and Constitution Avenue. The smaller ‘Parliamentary triangle’ is bounded by Commonwealth Avenue, Kings Avenue and King Edward Terrace. The axis of symmetry of the triangle runs from Parliament House, across Lake Burley Griffin, and directly along Anzac Parade to the Australian War Memorial. What other geometrical features can you see in Canberra’s design?

9780170194662

493

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Worksheet Congruent triangles proofs MAT10MGWK10079

13-02 Congruent triangle proofs Congruent figures are identical in shape and size. Matching sides are equal, and matching angles are equal.

Worksheet Congruent triangles

Summary

MAT10MGWK00022

There are four congruence tests for triangles: SSS, SAS, AAS or RHS. Two triangles are congruent if: • the three sides of one triangle are respectively equal to the three sides of the other triangle (SSS rule) • two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other triangle (SAS rule) • two angles and one side of one triangle are respectively equal to two angles and the matching side of the other triangle (AAS rule)

• they are right-angled and the hypotenuse and another side of one triangle are respectively equal to the hypotenuse and another side of the other triangle (RHS rule).

The congruence symbol ” The symbol for ‘is congruent to’ is a special equals sign, written as ‘”’ (which also means ‘is identical to’). When using this notation, we must make sure that the vertices (angles) of the congruent figures are written in matching order. For example, ‘nABC ” n XYZ’ means ‘triangle ABC is congruent to triangle XYZ’ and means \A ¼ \ X, \ B ¼ \Y, \C ¼ \ Z. C

Z

A

X B

494

Y

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

Example

ustralian Curriculum

10 þ10A

6 Video tutorial

D

In this diagram, \ DEF ¼ \ DFE and DG ’ EF.

Congruent triangles proofs

a Prove that n DEG ” n DFG. b Hence show that G is the midpoint of EF.

MAT10MGVT10019

E

F

G

Solution a In n DEG and n DFG: \ DEF ¼ \ DFE

Identifying the triangles in matching order of vertices.

(given)

\ DGE ¼ \DGF ¼ 90 DG is common. [ nDEG ” nDFG

Stating each part of the congruence test, giving reasons. (DG ’ EF)

(AAS)

Concluding the congruence proof, stating the test used.

b [ EG ¼ FG (matching sides of congruent triangles) [ G is the midpoint of EF.

Example

7

In the diagram, PQRT is a parallelogram. TR is extended to W so that TR ¼ RW. Prove that: a n PQV ” nWRV

T

R

W

b V is the midpoint of PW.

V

Solution a In n PQV and nWRV: PQ ¼ TR (opposite sides of a parallelogram are equal) TR ¼ RW (given) [ PQ ¼ RW \ Q ¼ \WRV (alternate angles, PQ || TW) \ PVQ ¼ \ WVR (vertically opposite angles) [ n PQV ” n WRV (AAS)

P

Q

b [ PV ¼ WV (matching sides of congruent triangles) [ V is the midpoint of PW.

9780170194662

495

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Exercise 13-02 Congruent triangle proofs See Example 6

A

1 AB ¼ CB and EB ¼ DB. Prove that n ABE ” n CBD.

D B

E

C

2 LM ¼ NP and LP ¼ NM. Prove that n LMP ” n NPM.

N

P

L

3 QT ’ WT, PW ’ WT and QW ¼ PT. Prove that n QTW ” nPWT.

M

P

W

T

4 ABCD is a square and AY ¼ CX. Prove that n ABY ” nCBX.

Q X

D

C

Y A

5 If \CDE ¼ \FED and DY ¼ EY in the diagram, prove that n CDE ” nFED.

D

B E

Y

C

F

6 O is the centre of both circles. Prove that n XOY ” n VOW.

Y V

O X

W

7 KLMN is a quadrilateral with opposite sides parallel. Prove that n KLM ” n MNK.

N

M

K

496

L 9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A

H

8 CH || EG, DH || FG and CH ¼ EG. Prove that nCDH ” nEFG.

G

C

D

E

F

U

9 YX bisects \UXW and UX ¼ WX. Prove that n UXY ” n WXY. X

Y

W

10 n ABC is isosceles with BA ¼ BC. AE ’ BC and CD ’ BA. Prove that n ABE ” n CBD.

C E

B D A H

11 \ HEF ¼ \ GFE and EH ¼ FG. Prove that: a nHEF ” n GFE

b \ EHF ¼ \ FGE. E

F B

12 O is the centre of the circle and AB ¼ CD. Prove that: a nAOB ” n COD

See Example 7

G K

b \ AOB ¼ \ COD.

D A

O

C Q

13 n QRT is isosceles with QR ¼ QT. If RX ¼ TY, prove that: a nQRX ” n QTY

b n QXY is isosceles.

R

14 TP ¼ XP and AP ¼ CP. Prove that:

X

T

Y

A X

a nTAP ” n XCP

P

b TA ¼ XC c TA || XC

9780170194662

T C

497

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

15 O is the centre of the circle and OM ’ AB. Prove that: a n OAM ” n OBM b OM bisects AB.

O

A

B

M G

16 GHKL is a kite with GL ¼ KL and GH ¼ KH. Prove that: a n GLH ” n KLH b LH bisects \GLK and \ GHK. L

H

K

Stage 5.3

13-03 Tests for quadrilaterals

Worksheet Quadrilaterals: True or false?

Summary

MAT10MGWK00020 Technology GeoGebra: Making quadrilaterals

Quadrilateral Trapezium

MAT10MGTC00012

498

Formal definition

Other properties

quadrilateral with at least one pair of opposite sides parallel

Parallelogram

quadrilateral with both pairs of opposite sides parallel

• opposite sides equal • opposite angles equal • diagonals bisect each other

Rhombus

parallelogram with two adjacent sides equal in length

• all sides equal • diagonals bisect each other at right angles • diagonals bisect the angles of the rhombus

Rectangle

parallelogram with a right angle

• • • •

opposite sides parallel/equal all angles are right angles diagonals are equal in length diagonals bisect each other

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

Quadrilateral Square

Kite

ustralian Curriculum

Formal definition rectangle with two adjacent sides equal in length

convex quadrilateral with two pairs of equal adjacent sides

10 þ10A

Other properties • all sides equal • all angles are right angles • diagonals are equal in length • diagonals bisect each other at right angles • diagonals bisect the angles of a square

Stage 5.3

• one pair of opposite angles equal • diagonals intersect at right angles

Some properties of the special quadrilaterals can be used as minimum conditions to prove or test that a quadrilateral is a parallelogram, rectangle, square or rhombus, for example, if opposite angles are equal, then it must be a parallelogram.

Summary Tests for quadrilaterals A quadrilateral is a parallelogram if one of the following is true: • • • • •

both pairs of opposite angles are equal, or both pairs of opposite sides are equal, or both pairs of opposite sides are parallel, or one pair of opposite sides are equal and parallel, or the diagonals bisect each other.

A quadrilateral is a rectangle if one of the following is true: • all angles are 90, or • diagonals are equal and bisect each other. A quadrilateral is a rhombus if one of the following is true: • all sides are equal, or • diagonals bisect each other at right angles. A quadrilateral is a square if one of the following is true: • all sides are equal and one angle is 90, or • all angles are 90 and two adjacent sides are equal, or • diagonals are equal and bisect each other at right angles.

9780170194662

499

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Stage 5.3

Example

8

ABCD is a quadrilateral with BD as a diagonal. Prove that if the opposite sides of ABCD are equal, then it must be a parallelogram.

D

Solution

C

B

A

In nABD and n CDB: AD ¼ CB (opposite sides of ABCD are equal) AB ¼ CD (opposite sides of ABCD are equal) BD is common. [ n ABD ” n CDB (SSS) [ \ ABD ¼ \ CDB (matching angles of congruent triangles) [ AB || CD (alternate angles are equal) Also, \ ADB ¼ \CBD (matching angles of congruent triangles) [ AD || CB (alternate angles are equal) [ ABCD is a parallelogram (opposite sides are parallel)

Example

9

In the diagram, KP || BM, AP || LM and KP ¼ BM. Prove that: a n KAP ” nBLM b ALMP is a parallelogram.

Solution

M

P

K

A B

L

a In n KAP and nBLM: \ PKA ¼ \ MBL (corresponding angles, KP || BM) \ KAP ¼ \ BLM (corresponding angles, AP || LM) KP ¼ BM (given) [ nKAP ” n BLM (AAS) b [ AP ¼ LM (matching sides of congruent triangles) AP || LM (given) [ ALMP is a parallelogram (one pair of opposite sides are parallel and equal)

500

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A Stage 5.3

Exercise 13-03 Tests for quadrilaterals 1 ABCD is a quadrilateral in which opposite angles are equal. Prove that ABCD must be a parallelogram.

D

× C

See Example 8

× A

2 LMNP is a quadrilateral in which LM ¼ NP and LM || NP. Prove that, if a pair of opposite sides in a quadrilateral are equal and parallel, then the quadrilateral must be a parallelogram.

B N

P

M

L

3 DEGH is a quadrilateral whose diagonals DG and EH bisect each other. Prove that it must be a parallelogram.

H

G X

D

E

4 CDEF is a quadrilateral whose diagonals CE and DF bisect each other at right angles. Prove that CDEF must be a rhombus.

F

E H C

5 VWXY is a quadrilateral whose diagonals VX and WY are equal and bisect each other. Prove that it must be a rectangle.

D

Y

X T

V

6 BCDE is a quadrilateral with all its angles equal to 90. Prove that its opposite sides are parallel and that hence it must be a rectangle.

7 TWME is a quadrilateral with all sides equal and \M ¼ 90. Prove that the other angles are 90 as well and that hence it must be a square.

9780170194662

W E

D

B

C

E

M

T

W

501

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Stage 5.3

8 GHKL is a quadrilateral with all angles equal to 90 and GH ¼ GL. Prove that all sides are equal and hence that it must be a square.

L

K

G

H

T

P

9 MNPT is a quadrilateral whose diagonals MP and NT are equal and bisect each other at right angles. Prove that MNPT must be a square.

X

M See Example 9

10 ABCD is a parallelogram and BX ¼ DY. Prove that:

B

N X

C

a n ABX ” nCDY b AXCY is a parallelogram.

A

Y

A

11 AECD is a rhombus and AE ¼ EB. Prove that:

D

E

B

a n DAE ” n CEB b BCDE is a parallelogram. D

12 ABCD is a parallelogram and AP ¼ AS ¼ CQ ¼ CR. Prove that: a RQ ¼ PS and PQ ¼ RS b PQRS is a parallelogram.

A

C

P

B

S

Q

D

13 AC and DB are diameters of concentric circles with centre O. Prove that ABCD is a parallelogram.

C

R A E B O

D F C

502

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

14 PR and SQ are diameters of concentric circles, centre O and TU ’ SQ. Prove that PQRS is a rhombus.

10 þ10A

T

Stage 5.3

P

S

Q

O

R U

15 DEFG is a rectangle. W, X, Y and Z are the midpoints of the sides. Prove that WXYZ is a rhombus.

D

W

E

Z

G

16 ABCD is a parallelogram. P, Q, R and T are the midpoints of the sides. Prove that PQRT is a parallelogram.

X

R

D

F

Y C

Q

T A

P

B

Investigation: Is a square a rhombus? Using the definitions of the special quadrilaterals, we see that a parallelogram can also be classified as a trapezium since it has at least one pair of opposite sides parallel. This means that trapeziums are inclusive of parallelograms. Similarly, parallelograms are inclusive of rectangles and rectangles are inclusive of squares. This can be represented by a Venn diagram. quadrilaterals trapeziums parallelograms rhombuses rectangles squares

1 Why is a rectangle a special type of parallelogram but a parallelogram is not always a rectangle? How can you use the Venn diagram to explain this? 2 Where would you put kites on the Venn diagram?

9780170194662

503

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Worksheet Proving properties of quadrilaterals MAT10MGWK10080

Animated example Geometric problems and proofs

Proving properties of triangles 13-04 and quadrilaterals The properties of triangles and quadrilaterals can be proved using the congruence tests.

Example

10

MAT10MGAE00008

Y

n WXY is an isosceles triangle with YW ¼ YX. T is the midpoint of WX. a Prove that n YTW ” n YTX. b Explain why \YWT ¼ \YXT. c What geometrical result about isosceles triangles does this prove?

Solution

W

T

X

a For n YTW and n YTX: YW ¼ YX (given) YT is common. WT ¼ XT (T is the midpoint of WX) [ nYTW ” n YTX (SSS) b \ YWT ¼ \ YXT because they are matching angles of congruent triangles. c The angles opposite the equal sides of an isosceles triangle are equal.

Example

11

ABCD is a rectangle.

D

C

A

B

a Prove that n ABD ” n BAC b Hence show that the diagonals of a rectangle are equal.

Solution a In n ABD and n BAC: AD ¼ BC (opposite sides of a rectangle) AB is common. \ DAB ¼ \ CBA ¼ 90 (angles in a rectangle) [ nABD ” n BAC (SAS) b BD ¼ AC (matching sides of congruent triangles) [ The diagonals of a rectangle are equal.

504

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A

Exercise 13-04 Proving properties of triangles and quadrilaterals 1

A

n ABC is an isosceles triangle with AB ¼ AC. D is the midpoint of BC. a Prove that n ABD ” n ACD. b Explain why \ADB ¼ \ ADC.

See Example 10

c Hence prove that AD ’ BC. B

2

D

n KMN is an isosceles triangle in which KM ¼ KN and KP ’ MN. a Prove that n KMP ” nKNP. b Prove that the perpendicular from vertex K to the side MN bisects that side.

K

M

3

4

5

N

P

D

ABCD is a rectangle. a Prove that n AXB ” n CXD. b Hence show that the diagonals of a rectangle bisect each other. DEFG is a parallelogram. a Prove that n DEG ” n FGE. b Draw the other diagonal DF and prove that n DGF ” n FED. c Hence prove that the opposite angles of a parallelogram are equal.

C

C X

A

B G

F

D

E

BCDE is a rhombus, so all of its sides are equal and opposite sides are parallel. a Prove that n BED ” n BCD.

E

D

See Example 11

b Hence prove that the diagonal BD bisects \EBC and \ EDC. B

6

LMNP is a parallelogram, so its opposite sides are parallel and equal. a Prove that n LXM ” n NXP. b Hence prove that the diagonals of a parallelogram bisect each other.

7

P

N X

L

M Y

UWXY is a rhombus. a Prove that n UAW ” n XAW. b Prove that n UAW ” n UAY. c Hence prove that the diagonals of a rhombus bisect each other at right angles.

9780170194662

C

X A

U

W

505

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

8

In n DFE, \ D ¼ \E and FX ’ DE.

D

a Prove that nDXF ” n EXF. b Hence prove that the sides opposite the equal angles in a triangle are equal.

X

F

E

9

nXYW is an equilateral triangle. A and B are the midpoints of the sides XW and YW respectively.

W

a Prove that nXBW ” n XBY. A

b Prove that nYAX ” n YAW. c Hence prove that \X ¼ \ Y ¼ \W ¼ 60.

B

X

Mental skills 13

Y

Maths without calculators

Time before and time after 1

Study this example. What is the time 10 hours and 15 minutes after 1850 hours? 1850 þ 10 hours ¼ 0450 (next day). Count: ‘1850, 1950, 2050, 2150, 2250, 2350, 0050, 0150, 0250, 0350, 0450’ 0450 þ 15 min ¼ 0450 hours þ 10 min þ 5 min ¼ 0500 þ 5 min ¼ 0505 (next day) OR: 10 minutes

1850

2

506

5 minutes

1900

0500

= 10 hours 15 minutes

0505

Now find the time of day: a c e g i k

3

10 hours

3 hours 20 minutes after 9:05 a.m. 4 hours 35 minutes after 6:15 p.m. 2 hours 45 minutes after 0325 8 hours 30 minutes after 12:40 a.m. 6 hours 25 minutes after 0435 9 hours 50 minutes after 2:30 p.m.

b d f h j l

5 hours 40 minutes after 7:30 p.m. 11 hours 10 minutes after 11:45 a.m. 7 hours 5 minutes after 1705 4 hours 55 minutes after 10:20 p.m. 2 hours 15 minutes after 2050 3 hours 10 minutes after 8:25 a.m.

Study this example. What is the time 8 hours and 45 minutes before 1115? 1115  8 hours ¼ 0315 Count back: ‘1115, 1015, 0915, 0815, 0715, 0615, 0515, 0415, 0315’ (or 11  8 ¼ 3). 0315  45 min ¼ 0315  15 min  30 min ¼ 0300  30 min ¼ 0230

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A

OR: 30 minutes

0230

4

8 hours

15 minutes

0300

1100

= 8 hours 45 minutes

1115

Now find the time of day: a c e g i

1 hour 15 minutes before 7:20 p.m. 3 hours 20 minutes before 3:30 p.m. 2 hours 10 minutes before 1455 5 hours 25 minutes before 4:15 a.m. 4 hours 20 minutes before 2005

b d f h j

4 hours 40 minutes before 11:20 a.m. 5 hours 35 minutes before 8:25 a.m. 3 hours 45 minutes before 0740 9 hours 30 minutes before 9:45 p.m. 2 hours 15 minutes before 0615

Stage 5.3

13-05 Formal geometrical proofs General geometrical results can be proved by writing a geometrical argument, where reasons are given at each step of the argument. This is called deductive geometry.

Example

12

Puzzle sheet Geometrical proofs order activity MAT10MGPS10081

In the diagram, AB || CD, KL ¼ DL and \LDP ¼ 115. Find the value of w, giving reasons.

K w°

A

C

B

D L

115° P

Solution \KDL ¼ 180  115 ¼ 65 ) \LKD ¼ 65 \AKD ¼ 115 ) w ¼ 115  65

ðangles on a straight lineÞ ðequal angles of isosceles 4 KLDÞ ðcorresponding angles, AB jj CDÞ

¼ 50

9780170194662

507

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Stage 5.3

Example

13 E

In the diagram, AC || ED, AE || BD, BE || CD and CB ¼ CD. Prove that n ABE is an isosceles triangle.

D

Solution A

\EBD ¼ \CDB

ðalternate angles, BE jj CDÞ

and \EBD ¼ \AEB

ðalternate angles, AE jj BDÞ

[ \ CDB ¼ \ AEB But \CDB ¼ \CBD

ðequal angles of isosceles 4CBDÞ

and \BAE ¼ \CBD

ðcorresponding angles, AE jj BDÞ

C

B

[ \ CDB ¼ \ BAE [ \ AEB ¼ \ BAE [ n ABE is an isosceles triangle.

(two equal angles)

Exercise 13-05 Formal geometrical proofs See Example 12

K

1 KL ¼ ML and MN ¼ MP. Find x, giving reasons.

P

x° L

M

N

C

2 CE || AB, CD ¼ BD and AC ¼ BC. Find m, giving reasons.

D

E m°

42° B

A

3 NK bisects \ HKL. Find the size of \ NHK, giving reasons.

H N 93° 147° L

K

4 BCDE is a rhombus with its diagonals intersecting at G. Find the value of x, giving reasons for each step.

E

x°

M D

G B

508

116° C

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

5 n ABC is an isosceles triangle, where AB ¼ AC and BC || ED. Prove that n ADE is an isosceles triangle.

10 þ10A C

Stage 5.3 See Example 13

D

A

E

6 PY bisects \XYW, PW bisects \TWY and YX || WT. Prove that \ YPW ¼ 90.

B

Y

X

P W

7 TWXZ is a parallelogram and TZ ¼ TY ¼ UX. a Prove that n TZY ” n XWU. b Hence prove that TUXY is a parallelogram.

T

Z

Y

T

X

U

8 MNPT is a square. W and Y are the midpoints of sides TP and MT. a Prove that n MNY ” n TMW. b Prove that MW ’ NY.

W W

T

Y

P

X

M

9 ABDE is a parallelogram and BC ¼ BD. Prove that \ AED ¼ 2\BCD.

N

E

D

A

B

10 A and B are the centres of two circles that intersect at C and D. a Prove that nADB ” n ACB. b Hence prove that n DXB ” nCXB and that DX ¼ CX.

C D

X

A

B

C

11 If AC ¼ BC and DC ¼ EC, prove that AB || DE.

B E C D A

9780170194662

509

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Stage 5.3

12 nUXY is an equilateral triangle and WX ¼ XU. Prove that \WUY is a right angle.

U

W

X

W

13 If WY || PQ, prove that the angle sum of n PQT is 180.

Y

P

14 CA bisects \ FAB and DA bisects \ HAB. Prove that \CAD ¼ 90.

Y

T

Q B

C

D

15 Prove that the exterior angle of a triangle is equal to the sum of the interior opposite angles (that is, prove that \CBD ¼ \CAB þ \BCA).

H

A

F C

A

16 AC is the diameter of a semicircle with centre O. B is a point on the semicircle. Let \ ABO ¼ x and \ CBO ¼ y. Prove that \ABC is a right angle.

B

D

B x° y°

A

O

C

13-06 Similar figures Similar figures have the same shape but are not necessarily the same size. When a figure is enlarged or reduced, a similar figure is created. The original figure is called the original, while the enlarged or reduced figure is called the image. The scale factor describes by how much a figure has been enlarged or reduced.

Summary Scale factor ¼

• •

510

image length original length

If the scale factor is greater than 1, then the image is an enlargement. If the scale factor is between 0 and 1, then the image is a reduction.

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

Example

ustralian Curriculum

10 þ10A

14

Find the scale factor for each pair of similar figures. a

b

45 mm 27 mm

20 mm

15 mm

20 mm

12 mm

Solution 15 20 3 ¼ 4

Image length Original length

a Scale factor ¼

45 27 5 ¼ 3

b Scale factor ¼

ðor

20 Þ 12

Image length Original length

The similarity symbol ||| The symbol for ‘is similar to’ is ‘|||’. As with congruence notation, we must make sure that the vertices (angles) of the similar figures are written in matching order.

Summary If two figures are similar, then:

Shutterstock.com/Elizaveta Shagliy

• the matching angles are equal • the matching sides are in the same ratio

9780170194662

511

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Example

15

Test whether each pair of figures are similar.

a

b 30 mm

65°

24 mm

65°

14 mm

26 mm

97°

25 mm

15 mm

20 mm

16 mm

107° 97° 20 mm

107° 12 mm

10 mm 20 mm

Solution a For the two quadrilaterals, matching angles are equal and the ratios of matching sides are equal.

20 ¼ 25 ¼ 30 ¼ 15 ¼ 5 16 20 24 12 4

[ The quadrilaterals are similar. b For the two rectangles, matching angles are equal (90) but the ratios of matching sides are not equal. [ The rectangles are not similar.

10 ¼ 1 but 14 ¼ 7 20 2 26 13

Exercise 13-06 Similar figures See Example 14

512

1

By measurement, find the scale factor for each pair of similar figures. a

b

c

d

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

2

Copy each figure onto graph paper and draw its image using the given scale factor. 2 a Scale factor ¼ 3 b Scale factor ¼ 3

3

For each pair of similar figures: i list all pairs of matching angles

10 þ10A

ii list all pairs of matching sides iii use the correct notation to write a similarity statement relating them. a

b F

P

S

G

L Y W

D

M

T

Y

B

4

Q T

Test whether each pair of figures are similar. a

See Example 15

b

36 6 9 × 8

24

18

27

12

×

c

d 24 28

42 28.5

9 15

36

19

15 25

f

e 25 18

15 30

9780170194662

513

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Skillsheet Finding sides in similar triangles MAT10MGSS10014

Worksheet Finding sides in similar figures MAT10MGWK10082

Finding unknown lengths 13-07 in similar figures Example

16

The two triangles are similar. Find the values of d and k.

27 mm k mm

42 mm

28 mm d mm

Puzzle sheet

44 mm

Similar triangles MAT10MGPS00025

Solution Since the triangles are similar, the ratios of matching sides are equal. d 42 k 28 ¼ ¼ 44 28 27 42 42 28 d¼ k¼ 3 44 3 27 28 42 ¼ 66 ¼ 18 OR Scale factor ¼ 28 ¼ 2 42 3 d ¼ 44 4

2 3

k ¼ 27 3

¼ 66

Example

2 3

¼ 18

17 L

n KLN ||| nPMN. Find the value of y.

M 18

y

N 15

K

9

P

Solution MP ¼ PN LK KN y 15 ¼ 18 24 15 3 18 y¼ 24 1 ¼ 11 4

514

Ratios of matching sides are equal. KN ¼ 9 þ 15 ¼ 24

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A

Exercise 13-07 Finding unknown lengths in similar figures 1

Find the value of every pronumeral in each pair of similar figures. a

See Example 16

b 16 mm

w mm

20 mm

m cm

15 cm

28 mm 18 cm

27 cm

c

d

p mm

x mm

45 cm

12 mm h mm

25 mm

15 mm

35 mm

f

8 cm

20 cm q cm

a cm 10 cm

27 cm

w cm

g

h

16 mm 16 mm •

+

+

10 cm

g cm

16 cm

6 cm

20 mm

15 cm

12 mm + • b mm

8 cm

8 cm

11 cm

5 cm

+

e

30 mm

12 mm

t cm

u cm

y mm

2

A

n ABC ||| n ADE. Find the value of h.

See Example 17

7 E

D

8

5

C

B

3

h

n MNP ||| n MWY. Find the value of x.

12 P 15 M

9780170194662

Y 16

x N

W

515

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

This photograph of the Sydney Harbour Bridge has been enlarged so that its length is 24 cm. If the dimensions of the original photo were 15 cm 3 10 cm, what is the width of the enlargement?

10 cm Shutterstock.com/clearviewstock

4

15 cm

5

A building that is 40 m high casts a shadow 15 m long. At the same time, the shadow of a tree is 4.5 m long. What is the height of the tree?

40 m

15 m

6

7

nWXY ||| n WDE. What is the value of x? Select the correct answer A, B, C or D. A 4 B 6 C 8 D 10

4.5 m W 10

x

D

X

15

E Y

24

Katrina is 1.72 m tall and casts a shadow that is 2.5 m long. At the same time, a flagpole casts a shadow that is 3.5 m long. How long is the flagpole? 1.72 m 3.5 m

8

Which two rectangles are similar? Select the correct answer A, B, C or D.

K

A M and N 9

516

2.5 m

M

B K and P

N

C M and P

P

D K and N

A 2 m high fence casts a shadow of 1.4 m. How long is the shadow cast by a pole that is 3.2 m high at the same time?

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

ustralian Curriculum

10 þ10A

13-08 Tests for similar triangles There are four sets of conditions that can be used to determine if two triangles are similar. These are called the tests for similar triangles or similarity tests.

Summary There are four tests for similar triangles. Two triangles are similar if: • the three sides of one triangle are proportional to the three sides of the other triangle (‘SSS’) F

C

10

5

4

2 A

B

4

D

E

8

• two sides of one triangle are proportional to two sides of the other triangle, and the included angles are equal (‘SAS’) F 50

C 5

A

3

B

E 30

D

• two angles of one triangle are equal to two angles of the other triangle (‘AA’ or ‘equiangular’) C

F equiangular means ‘equal angles’ E

A B

D

• they are right-angled and the hypotenuse and a second side of one triangle are proportional to the hypotenuse and a second side of the other triangle (‘RHS’).

6

5 15 2

9780170194662

517

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Example

18

Which test can be used to prove that each pair of triangles are similar? a

9

5.4 108°

10

108°

15

b 6

12

20 9

c

8

d 35° 9

21

21°

14

12

35°

21° 15.75

Solution 10 9 5 a Two pairs of matching sides are in the same ratio ¼ ¼ and the included angles in 6 5:4 3 both are 108. (‘SAS’) b In both right-angled triangles, the pairs of hypotenuses and second sides are in the same ratio 12 ¼ 9 ¼ 3. (‘RHS’) 20 12 4 c All three pairs of matching sides are in the same ratio 8 ¼ 12 ¼ 9 ¼ 4. (‘SSS’) 14 21 15:75 7 d Two pairs of angles are equal. (‘AA’)

Exercise 13-08 Tests for similar triangles See Example 18

1

Which test can be used to prove that each pair of triangles are similar? a

b 56° 72°

22

15.5 11

72°

9

18

31

56° c

d

16

59°

34°

8

59° 34°

6 12 e

f

9 14.25

9

12

19

15

12

26 20.8

12

518

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

g

ustralian Curriculum

h

16

14.4

67°

67°

10 þ10A

27.5

18

22 20

i

j

10 23

81° 26

10

8

30 18.2

8

81°

13 13

21

6

2

For each set of triangles, find the pair of similar triangles.

a 24

47° B 12

A 47°

b

5 7 A 8.5

16 C 9 12 47°

47°

14

18

D

20

17

14

B

7.5

10

C

10

14

8

D 11.5

10.5

c 32 24 A

36

31.5

B

16

C

D

20 28

18

3

Use the correct notation to write a similarity statement relating each pair of similar triangles.

a

A

b H

U

K

10.5

D

L

10 13 52°

W

52°

8

20

15

10.4

M

Y

d

Q C

T

25°

128°

A

9780170194662

21

25°

12.5

11

B G

W

S

H P

128°

T

P

E

c

14

15.4

17.5

15

N

V

519

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

Stage 5.3

Worksheet Congruent and similar triangle proofs MAT10MGWK10083

Worksheet Complete the proofs MAT10MGWK10227

13-09 Similar triangle proofs To formally prove that two triangles are similar, we use a specific format that involves applying one of the four similarity tests: ‘SSS’, ‘SAS’, ‘AA’, ‘RHS’.

Example

19

Prove that each pair of triangles are similar. a

b

C

15

T

8

A

G

B

13.5

W K

58°

58° 10

11.25

9.8

7

18 H 6

Y

P X

14 R

Solution a In n ABC and nKHG: AB ¼ 18 ¼ 4 KH 13:5 3 AC ¼ 15 ¼ 4 KG 11:25 3 BC 8 4 ¼ ¼ HG 6 3 AB AC BC ¼ ¼ KH KG HG [ nABC ||| nKHG

(three pairs of matching sides are in proportion, or ‘SSS’)

b In n WXY and n PRT: WX ¼ 10 ¼ 5 PR 14 7 WY ¼ 7 ¼ 5 PT 9:8 7 WX ) ¼ WY PR PT \ W ¼ \ P ¼ 58 [ nWXY ||| n PRT (two pairs of matching sides are in proportion and the included angles are equal, or ‘SAS’)

520

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

Example

ustralian Curriculum

10 þ10A Stage 5.3

20 A

Prove that n ABC ||| n EDC and hence find the value of m.

m

Solution

D

10

In n ABC and n EDC: \A ¼ \E

ðalternate angles, AB jj DEÞ

\B ¼ \D

ðalternate angles, AB jj DEÞ

[ n ABC ||| n EDC m 16 ) ¼ 8 10 16 38 m¼ 10 ¼ 12:8

(equiangular, or ‘AA’)

C 8

16

B

E

ðmatching sides in similar trianglesÞ

Exercise 13-09 Similar triangle proofs 1

Prove that each pair of triangles are similar. a

b T

18

P 25

10.8

c C

W

B

d

25.5

W

T

123°

123°

22°

G 8

L

E

20

L

N 12

21

18

24 17

Q

12

E 22°

16

12 A

V

15

M

H

C

B

D M

2

See Example 19

V

a D and E are the midpoints of AB and AC. Prove that nADE ||| n ABC.

H

35

b AC || FD and BF || CE. Prove that n ABF ||| n FDE.

A

D

E

E F

B

9780170194662

See Example 20

C

A

B

D

C

521

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Geometry

c Prove that n WXY ||| n TXW.

Stage 5.3

d Prove that n NDL ||| n NQR.

Y

N 8

10

D L 12

X

e HW || XY. Prove that nXWH ||| n YXW. 18

H

Q

R

T

W

6

f NCKL is a parallelogram. Prove that n NML ||| n KLP. W

L

K

12 P X

Y

8

N

3

M

C

b i Prove that n ACE ||| n BCD. ii Find the value of y.

a i Prove that nFLN ||| n FDE. ii Find the value of d.

E

F

D

12 8.5 L

d

5.5 N

8

y

A

D

10

B

E

15

c i Prove that nYRT ||| n WUT. ii Find the value of g.

d i Prove that n NMP ||| n PCB. ii Find the value of w.

Y

N w

20

R

6 g

T

9

P

U

M

5

8 W

522

C

6 B

C

T

9780170194662

N E W C E N T U R Y M AT H S A D V A N C E D for the A

e i Prove that n TYN ||| n YNM. ii Find the value of h. T

9

ustralian Curriculum

10 þ10A

f i Prove that nBHU ||| n XBD. ii Find the value of y. 18

U

Y

Stage 5.3

H 9

X

h y N

4

16

M

B

12

D

GHLM is a rectangle and K is the midpoint of HL. a Prove that nMXG ||| n KXL. b Find the value of x.

L

M x X

7 K

8

G

5

H

a Prove that nCLW ||| n LTE.

W

T

b If WT ¼ 5 cm, CE ¼ 15 cm and EL ¼ 6 cm, find the length of TL.

L E

C

6

PTUK is a rectangle and KB ’ PU. a Prove that nPTU ||| nKBP.

U

K

b If BU ¼ 21 cm and KP ¼ 10 cm, find the length of PB. B T

P

Power plus 1

a Prove that the three triangles ABC, ACD and CBD are similar. b Find the length of CD.

C 12 cm

A

2

3

5 cm

13 cm

D

B

ABCD is a parallelogram. HG is any interval joining parallel sides AB and DC and passing through the midpoint of the diagonal BD. Prove that the interval through the midpoint of a diagonal in a parallelogram divides opposite sides equally (that is, prove DG ¼ BH). B The median is a line joining a vertex of a triangle to the midpoint of the opposite side. Prove that the medians of a triangle are concurrent (that is, they meet at one intersection point). A C

9780170194662

523

Chapter 13 review

n Language of maths AAS

angle sum

congruence test

congruent (”)

Geometry crossword

convex polygon

enlargement

equiangular

exterior angle

MAT10MGPS10084

hypotenuse

image

included angle

matching

original

polygon

proof

proportional

quadrilateral test

regular polygon

RHS

SAS

scale factor

similar (|||)

similarity test

SSS

Puzzle sheet

1 What is a convex polygon? 2 Explain the difference between the interior and exterior angles of a polygon. 3 What is the symbol and meaning of ‘is similar to’? 4 What happens to a figure that is changed by a scale factor of 1? 2 5 What are the four tests for similar triangles? 6 What is the meaning of the ‘A’ in the SAS test for congruent triangles? 7 What does equiangular mean in the similarity tests?

n Topic overview Quiz Geometry

Copy and complete this mind map of the topic, adding detail to its branches and using pictures, symbols and colour where needed. Ask your teacher to check your work.

MAT10MGQZ00008

Congruent triangle proofs

Angle sum of a polygon

Similar triangle proofs

524

Proving properties of triangles and quadrilaterals

GEOMETRY

Finding unknown sides in similar figures

Formal geometrical proofs

Similar figures

9780170194662

Chapter 13 revision 1 Find the size of one angle in a regular 15-sided polygon.

See Exercise 13-01

2 The angle sum of a polygon is 6120. How many sides does the polygon have?

See Exercise 13-01

3 Find the number of sides in a regular polygon if each exterior angle is:

See Exercise 13-01

a 10

b 24

c 45

d 15

4 Each angle in a regular polygon is 162. How many sides does the polygon have? Select the correct answer A, B, C or D. A 18

B 20

C 22

See Exercise 13-01

D 24 Y

5 In n WXY, \ W ¼ \ X and YZ ’ WX. Prove that n WZY ” n XZY.

W

X

Z D

6 ABCD is a parallelogram and BC ¼ BY ¼ DX. a Prove that n DAX ” nBCY. b Hence, prove that BXDY is a parallelogram.

See Exercise 13-02

Y

C

Stage 5.3 See Exercise 13-03

A

7 PNML is a rectangle. a Prove that n PML ” n NLM. b Hence explain why PM ¼ NL. c What geometrical result about rectangles does this prove?

X

B N

P

See Exercise 13-04

T L

M N

8 PNMQ is a square and AM ¼ BQ. Prove that n NPC is isosceles.

M

Stage 5.3 See Exercise 13-05

A C B P

Q

9 Test whether each pair of figures are similar. a

See Exercise 13-06

b 11.25 15 15 27

10

22

16

18

27.5 9

9780170194662

20

12

525

Chapter 13 revision See Exercise 13-07

10 If n ABE ||| n ACD, find the value of d.

A

7 cm

E

9 cm

D

5 cm B

d cm

C See Exercise 13-08

11 Which test can be used to prove that each pair of triangles are similar? a

13.5

b

18

47°

18

15 47°

16 23 30

20 10

c

22°

22°

121°

121°

Stage 5.3 See Exercise 13-09

526

12 Use similar triangles to prove that the interval joining the midpoint of two sides in a triangle is parallel to the third side and is half its length.

9780170194662