Similarity and Enlargement

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

1. Question 1

   4 marks

   Triangle PQR has vertices P(1, 1), Q(4, 1) and R(1, 3). It is mapped onto triangle P′Q′R′ by an enlargement with centre (0, 0) and linear scale factor 3. State the coordinates of P′, Q′ and R′, and state the size of angle Q′P′R′ given that angle QPR is 90°.

   (4 marks)

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2. Question 2

   1 mark

   Identify the pair of triangles that must always be similar.

   A. Two triangles with two pairs of equal sides

   B. Two triangles with two pairs of equal angles

   C. Any two right-angled triangles

   D. Any two isosceles triangles

   (1 mark)

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3. Question 3

   5 marks

   Two flower beds at a school are mathematically similar rectangles. Their areas are 4 m² and 9 m², and the shorter side of the smaller bed is 1.6 m. Determine the shorter side of the larger bed and the ratio of the perimeter of the smaller bed to that of the larger bed.

   (5 marks)

   Marking scheme

   Award 1 mark: area scale factor = $\frac{9}{4}$. Award 1 mark: linear scale factor = $\sqrt{\frac{9}{4}}=\frac{3}{2}$. Award 1 mark: shorter side of larger $bed=1.6\times \frac{3}{2}$. Award 1 mark: = 2.4 m. Award 1 mark: perimeters are lengths, so they are in the linear ratio, giving 2 : 3. Note: a candidate who uses $\frac{9}{4}$ as the linear scale factor (giving 3.6 m) loses the second, third and fourth marks but may still earn the final mark for a correctly reasoned perimeter ratio.

4. Question 4

   1 mark

   Two similar triangular maize plots have corresponding sides of 6 m and 15 m. State the ratio of the area of the smaller plot to the area of the larger plot.

   A. 25 : 4

   B. 4 : 25

   C. 8 : 125

   D. 2 : 5

   (1 mark)

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5. Question 5

   3 marks

   A design on a school gate is enlarged with a linear scale factor of 3. State two properties of the design that remain unchanged and one property that changes, giving the effect of the change.

   (3 marks)

   Marking scheme

   Award 1 mark: the sizes of all the angles remain unchanged. Award 1 mark: the shape remains the same, that is the ratio of any two sides within the figure is unchanged (also accept: parallel lines remain parallel). Award 1 mark: the lengths change — every length in the image is 3 times the corresponding length in the object (accept instead: the area changes, becoming 9 times as large). Note: the third mark requires the size of the change, not just the word "bigger".

6. Question 6

   1 mark

   Late one afternoon on a school field in Machakos, a learner 1.5 m tall casts a shadow 2 m long while a nearby flagpole casts a shadow 12 m long on the same level ground. State the height of the flagpole.

   A. 18 m

   B. 11.5 m

   C. 9 m

   D. 16 m

   (1 mark)

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7. Question 7

   4 marks

   Two sufurias used in a school kitchen are mathematically similar. Their heights are 15 cm and 25 cm, and the smaller one holds 1.8 litres when full. Determine the capacity of the larger sufuria, giving the answer to two decimal places.

   (4 marks)

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8. Question 8

   1 mark

   A figure is enlarged about a fixed centre. State what happens to the centre of enlargement itself.

   A. It remains in the same position

   B. It shifts to the middle of the image

   C. It moves only when the scale factor is negative

   D. It moves away from the object by the scale factor

   (1 mark)

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9. Question 9

   4 marks

   In triangle ABC, point D lies on AB and point E lies on AC such that DE is parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm. Determine the length of BC.

   (4 marks)

   Marking scheme

   Award 1 mark: triangles ADE and ABC identified as similar, because DE parallel to BC makes angle ADE = angle ABC and angle AED = angle ACB (corresponding angles), with angle A common. Award 1 mark: AB = AD + DB = 4 + 6 = 10 cm. Award 1 mark: ratio of corresponding sides AD : AB = 4 : 10 = 2 : 5, so DE : BC = 2 : 5. Award 1 mark: $BC=5\times \frac{5}{2}=12.5$ cm. Note: a candidate who uses AD : DB = 4 : 6 loses the third and fourth marks, as DB is not a side of either triangle.

10. Question 10

    2 marks

    State two conditions that must both be satisfied for two polygons to be similar.

    (2 marks)

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11. Question 11

    1 mark

    Two similar cylindrical milk cans used at a dairy cooling plant have heights 12 cm and 18 cm. The base radius of the smaller can is 4 cm. State the base radius of the larger can.

    A. 10 cm

    B. 8 cm

    C. 2.7 cm

    D. 6 cm

    (1 mark)

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12. Question 12

    1 mark

    Identify the set of figures in which any two members are always similar to each other.

    A. All rhombuses

    B. All rectangles

    C. All parallelograms

    D. All circles

    (1 mark)

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13. Question 13

    1 mark

    A shape is mapped onto its image by an enlargement with centre O and linear scale factor −2. Identify the correct description of the image.

    A. Twice as long as the object, inverted, and on the opposite side of O

    B. The same size as the object but on the opposite side of O

    C. Half as long as the object, inverted, and on the opposite side of O

    D. Twice as long as the object, the same way up, and on the same side of O

    (1 mark)

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14. Question 14

    2 marks

    Distinguish between two figures that are similar and two figures that are congruent.

    (2 marks)

    Marking scheme

    Award 1 mark: similar figures have equal corresponding angles and corresponding sides in a constant ratio, so their sizes may differ. Award 1 mark: congruent figures have equal corresponding angles and corresponding sides equal in length (ratio 1 : 1), so they are identical in both shape and size; congruent figures are therefore a special case of similar figures. Note: "congruent means the same" without reference to sides or the 1 : 1 ratio earns 0 marks for that point.

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