12. Similar Shapes (Basics)

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

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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 $60\degree$, 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 $45\degree$ 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 $1.4\times$ the side lengths in shape D.

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

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

Marks = 1

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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.

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

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 $EF = 12\times 3 = 36$

Marks = 1

 

3(c):

Hence, or otherwise, calculate length AC in cm

ANSWER: Simple Text Answer

Answer: 17

Workings:

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

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

Marks = 1

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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: $\dfrac{3}{2}$

Workings:

$AB$ and $CD$ are matching sides

Dividing $CD$ by $AB$ gives the scale factor

$\dfrac{18}{12} = \dfrac{3}{2}$

Marks = 1

 

4(b):

Using the scale factor, calculate XC in cm

ANSWER: Simple Text Answer

Answer: 21

Workings:

$XC = \dfrac{3}{2} \times 14$

$XC = 21$

Marks = 1

 

4(c):

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

ANSWER: Multiple Answers (Type 1)

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

Workings:

$AD = AX + XD = AX + \dfrac{3}{2}AX$

$AD = \dfrac{5}{2}AX$

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

$XD = AD - AX = 25 - 10 = 15$

Marks = 2

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Question 5

Triangles $BCA$ and $BED$ are mathematically similar.

$BC=4.4 cm$

$BA=3 cm$

$AD=3 cm$

$AC=5 cm$

5(a):

What is the scale factor that transforms ABC to BDE?

ANSWER: Simple Text Answer

Answer: 2

Workings:

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

Marks = 1

 

5(b):

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

ANSWER: Multiple Answers (Type 1)

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

Workings:

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

$BE = 4.4\times 2 = 8.8$

Marks = 2

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Question 6

ABCD and AGFE are mathematically similar

$EF=48 cm$ $CB=16 cm$

6(a):

What is the scale factor of ADCB to AEFG?

ANSWER: Simple Text Answer

Answer: 3

Workings:

$Scale Factor = \dfrac{EF}{CB} = \dfrac{48}{16} = 3$

Marks = 1

 

6(b):

The area of ABDC is 24 $cm^2$.

Calculate the area of AEFG in $cm^2$.

ANSWER: Simple Text Answer

Answer: 216

Workings:

Scale Factor for area is Scale Factor squared for side length

$3^2 = 9$

$9\times 24 = 216$

Marks = 2

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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 $HI$ is an enlargement of $IJ$ and that $HM$ is an enlargement of $JK$

$HI$ and $JK$ are the same length so let $HI = JK = x$

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

Rearranging gives $x^2 = 100$

$x = 10$

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

Marks = 3