Similarity and Enlargement

14 Grade 10 Essential Mathematics questions on Similarity and Enlargement, with worked marking schemes. 14 questions are available on this topic in full.

Question 1
4 marks
Triangle ABC has vertices A(1, 1), B(3, 1) and C(1, 4). It is enlarged with centre the origin and scale factor 3. State the coordinates of the images A′, B′ and C′, and state the ratio of the area of triangle ABC to the area of triangle A′B′C′.
(4 marks)
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Question 2
1 mark
At the same time of day, a learner 1.5 m tall casts a shadow 2 m long while the school flagpole casts a shadow 8 m long. The height of the flagpole is:
A. 4 m
B. 12 m
C. 6 m
D. 10.7 m
(1 mark)
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Question 3
3 marks
Triangle ABC is similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. Given that AB = 6 cm, BC = 9 cm and PQ = 8 cm, calculate the length of QR.
(3 marks)
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Question 4
1 mark
Which one of the following statements is always true?
A. All squares are similar
B. All rectangles are similar
C. All rhombuses are similar
D. All isosceles triangles are similar
(1 mark)
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Question 5
6 marks
A carpenter builds a scale model of a grain store using a scale of 1 : 25. The model is 60 cm high, the outside surface to be painted measures 10 800 cm² and the model encloses a volume of 72 000 cm³. Calculate (a) the actual height of the grain store in metres, (b) the actual surface area to be painted in square metres, and (c) the actual volume of the store in cubic metres.
(6 marks)
Marking scheme
Award (a): $60 \times 25 = 1500$ cm = 15 m (1). Award (b): area scale factor $2 5^{2} = 625$ stated (1); $10800 \times 625 = 6750000$ cm² (1); converted using $1 m^{2} = 10000$ cm² to give 675 m² (1). Award (c): volume scale factor $2 5^{3} = 15625$ stated (1); $72000 \times 15625 = 1125000000$ cm³, converted using $1 m^{3} = 1000000$ cm³ to give 1 125 m³ (1).
Question 6
1 mark
The point P(2, 3) is mapped onto P′ by an enlargement with centre the origin and scale factor −2. The coordinates of P′ are:
A. (−4, 6)
B. (4, 6)
C. (−4, −6)
D. (−1, −1.5)
(1 mark)
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Question 7
1 mark
A photograph 6 cm wide is enlarged so that the width of the copy is 15 cm. The linear scale factor of the enlargement is:
A. 0.4
B. 9
C. 2.5
D. 2
(1 mark)
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Question 8
1 mark
When a figure is enlarged with scale factor 3, which property of the figure remains unchanged?
A. The lengths of the sides
B. The sizes of the angles
C. The area
D. The perimeter
(1 mark)
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Question 9
2 marks
Distinguish between two similar figures and two congruent figures.
(2 marks)
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Question 10
3 marks
In triangle ABC, D is a point on AB and E is a point on AC such that DE is parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm.
(a) Give a reason why triangle ADE is similar to triangle ABC.
(b) Calculate the length of BC.
(3 marks)
Marking scheme
Award (a): because DE is parallel to BC, angle ADE = angle ABC and angle AED = angle ACB (corresponding angles), and angle A is common, so the two triangles are equiangular (1). Award (b): AB = 4 + 6 = 10 cm and the ratio statement AD/AB = DE/BC, i.e. $\frac{4}{10} = 5/ B C$ (1). $B C = (5 \times 10) \div 4 = 12.5$ cm, with the unit (1).
Question 11
1 mark
Two similar triangular maize plots have corresponding sides in the ratio 3 : 5. The ratio of their areas is:
A. 3 : 5
B. 9 : 25
C. 27 : 125
D. 15 : 25
(1 mark)
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Question 12
1 mark
Two similar plastic jerricans used for fetching water have heights 20 cm and 40 cm. If the smaller jerrican holds 3 litres when full, the capacity of the larger one is:
A. 12 litres
B. 6 litres
C. 24 litres
D. 48 litres
(1 mark)
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Question 13
5 marks
Two similar cylindrical water tanks stand outside a school kitchen. The larger tank is 2.4 m high and the smaller one is 1.6 m high. The smaller tank holds 1 350 litres when full.
(a) Calculate the capacity of the larger tank.
(b) Explain why the capacity of the larger tank is not simply 1.5 times that of the smaller one.
(5 marks)
Marking scheme
Award (a): linear scale factor $2.4 \div 1.6 = 1.5$ (1). Volume scale factor $(1.5)^{3} = 3.375$ (1). Capacity = $1350 \times 3.375$ (1). Capacity = 4 556.25 litres (1). Award (b): explanation — capacity is a volume, so all three dimensions (height, and both across the base) are multiplied by 1.5; the volumes therefore compare in the ratio of the cube of the linear scale factor, not the scale factor itself (1).
Question 14
4 marks
Describe the effect on an object when it is enlarged about a fixed point with a scale factor of − $\frac{1}{2}$ .
(4 marks)
Marking scheme
Award: the image lies on the opposite side of the centre of enlargement from the object (1). The image appears inverted, as though the object has been turned through 180° about the centre (1). Every length of the image is half the corresponding length of the object (1). The angles are unchanged, so the image is similar to the object (accept: the image is smaller than the object) (1).
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More Grade 10 Essential Mathematics topics

Similarity and Enlargement

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14