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\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
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
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
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
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
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