Question 1

1(a):

Choose the correct definition for the term ‘similar shape’.

ANSWER: Multiple Choice (Type 1)

A: Similar shapes have the same number of sides

B: Similar shapes have sides all the same length

C: Similar shapes are shapes that have been rotated

D: Similar shapes are enlargements of each other

Answer: D

Workings:

Similar shapes have the same sized angles but side lengths that have changed by the same scale amount.

Marks = 1

 

1(b):

Using the shapes above, find a triplet of similar shapes

ANSWER: Multiple Choice (Type 1)

A: C, A, D

B: A, G, F

C: J, K, L

D: C, H, B

Answer: C

Workings:

Shapes J, K and L are all circles, but are of different sizes so must be enlargements of each other. There are no angles in a circle so they must be similar shapes.

Marks = 1


Question 2

The diagram below shows a variety of triangles.

In the set of shapes there are two pairs of similar triangles and one triplet of similar triangles.

2(a):

From the shapes above identify which of the following is a pair of similar triangles

ANSWER: Multiple Choice (Type 1)

A: B & G

B: A & F

C: D & E

D: C & B

Answer: B

Workings:

From the information given, we know shapes A and F are equilateral triangles.

This must mean that all their angles are 60°60\degree, so regardless of the side lengths, they are similar shapes.

Marks = 1

 

2(b):

From the shapes above identify which of the following is a pair of similar triangles

ANSWER: Multiple Choice (Type 1)

A: G & H

B: A & D

C: C & F

D: B & E

Answer: D

Workings:

Shapes B and E are both isosceles with a right angle, so the base angles must be 45°45\degree in both diagrams.

This means that, regardless of their side lengths, they are similar shapes.

Marks = 1

 

2(c):

From the shapes above identify which of the following is a triplet of similar triangles

ANSWER: Multiple Choice (Type 1)

A: E, D, F

B: C, D, H

C: A, B, H

D: B, F, G

Answer: B

Workings:

Shapes C, D and H are all right angled triangles.

We are given corresponding lengths for each side that are increased by the same scale factor.

The sides in shape C are 1.4×1.4\times the side lengths in shape D.

The sides in shape H are 1.6×1.6\times the side lengths in shape D.

Therefore the angles must stay the same and the shapes are similar.

Marks = 1


Question 3

The two triangles, ABC and DEF shown below are similar.

3(a):

What is the scale factor that transforms ABC to DEF?

ANSWER: Simple Text Answer

Answer: 3

Workings:

Find the scale factor by dividing the 42 cm side in the right-hand shape by the 14 cm side in the left-hand shape as they are corresponding sides.

4214=3\dfrac{42}{14} = 3

Marks = 1

 

3(b):

Hence, or otherwise, calculate length EF in cm

ANSWER: Simple Text Answer

Answer: 36

Workings:

The corresponding side in the left-hand shape is 12 cm

Increasing by scale factor 3 means EF=12×3=36EF = 12\times 3 = 36

Marks = 1

 

3(c):

Hence, or otherwise, calculate length AC in cm

ANSWER: Simple Text Answer

Answer: 17

Workings:

Use scale factor=13scale  factor = \dfrac{1}{3} to get from the right-hand shape to the left-hand shape.

513=17\dfrac{51}{3} = 17

Marks = 1


Question 4

In the diagram below the triangles XCD and XAB are similar.

4(a):

What is the scale factor of the triangles?

ANSWER: Simple Text Answer

Answer: 32\dfrac{3}{2}

Workings:

ABAB and CDCD are matching sides

Dividing CDCD by ABAB gives the scale factor

1812=32\dfrac{18}{12} = \dfrac{3}{2}

Marks = 1

 

4(b):

Using the scale factor, calculate XC in cm

ANSWER: Simple Text Answer

Answer: 21

Workings:

XC=32×14XC = \dfrac{3}{2} \times 14

XC=21XC = 21

Marks = 1

 

4(c):

Given that AD=25cmAD = 25cm, find the values of AX and XD in cm.

ANSWER: Multiple Answers (Type 1)

Answer: AX=10AX = 10, XD=15XD = 15

Workings:

AD=AX+XD=AX+32AXAD = AX + XD = AX + \dfrac{3}{2}AX

AD=52AXAD = \dfrac{5}{2}AX

AX=25(52)=10AX = \dfrac{25}{(\dfrac{5}{2})} = 10

XD=ADAX=2510=15XD = AD - AX = 25 - 10 = 15

Marks = 2


Question 5

Triangles BCABCA and BEDBED are mathematically similar.

BC=4.4cmBC=4.4 cm

BA=3cmBA=3 cm

AD=3cmAD=3 cm

AC=5cmAC=5 cm

5(a):

What is the scale factor that transforms ABC to BDE?

ANSWER: Simple Text Answer

Answer: 2

Workings:

Scale Factor=BDAB=63=2Scale  Factor = \dfrac{BD}{AB} = \dfrac{6}{3} = 2

Marks = 1

 

5(b):

Use the scale factor to calculate the length of BE and DE.

ANSWER: Multiple Answers (Type 1)

Answer: BE=8.8BE = 8.8, DE=10DE = 10

Workings:

DE=AC×2=5×2=10DE = AC\times 2 = 5\times 2 = 10

BE=4.4×2=8.8BE = 4.4\times 2 = 8.8

Marks = 2


Question 6

ABCD and AGFE are mathematically similar

EF=48cmEF=48 cm CB=16cmCB=16 cm

6(a):

What is the scale factor of ADCB to AEFG?

ANSWER: Simple Text Answer

Answer: 3

Workings:

ScaleFactor=EFCB=4816=3Scale Factor = \dfrac{EF}{CB} = \dfrac{48}{16} = 3

Marks = 1

 

6(b):

The area of ABDC is 24 cm2cm^2.

Calculate the area of AEFG in cm2cm^2.

ANSWER: Simple Text Answer

Answer: 216

Workings:

Scale Factor for area is Scale Factor squared for side length

32=93^2 = 9

9×24=2169\times 24 = 216

Marks = 2


Question 7

In the diagram below, IJKH and ILMH are mathematically similar.

IJ=5 cm

HM=20 cm

Find the scale factor of  IJKH to ILMH.

ANSWER: Simple Text Answer

Answer: 2

Workings:

Recognise HIHI is an enlargement of IJIJ and that HMHM is an enlargement of JKJK

HIHI and JKJK are the same length so let HI=JK=xHI = JK = x

x5=20x\dfrac{x}{5} = \dfrac{20}{x}

Rearranging gives x2=100x^2 = 100

x=10x = 10

Substituting this gives 105=2\dfrac{10}{5} = 2 as the Scale Factor

Marks = 3