Similarity & Enlargement

11 Grade 9 Mathematics questions on Similarity & Enlargement, with worked marking schemes. 11 questions are available on this topic in full.

1. Question 1

   1 mark

   Two similar triangles have areas 16 cm² and 100 cm². The ratio of their corresponding sides is ___

   A. 4:10

   B. 1:6

   C. 2:5

   D. 16:100

   (1 mark)

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2. Question 2

   1 mark

   A photo is enlarged by scale factor 3. If the original width is 4 cm, the new width is ___

   A. 1.33 cm

   B. 9 cm

   C. 12 cm

   D. 7 cm

   (1 mark)

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3. Question 3

   3 marks

   A map has scale 1:50,000. Two towns are 3 cm apart on the map. What is the actual distance in km?

   (3 marks)

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4. Question 4

   1 mark

   If the scale factor is 2, the area of the enlarged shape is ___ times the original area.

   A. 6

   B. 8

   C. 2

   D. 4

   (1 mark)

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5. Question 5

   2 marks

   A rectangular Kenyan flag measures $45$ cm long and $30$ cm wide. A tailor makes a larger, similar flag whose length is $75$ cm. Find the width of the larger flag.

   (2 marks)

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6. Question 6

   4 marks

   Triangle ABC has sides AB = 6, BC = 8, CA = 10. Triangle PQR is similar to ABC with PQ = 9. Find QR and PR.

   (4 marks)

   Marking scheme

   In similar triangles corresponding sides are in the same ratio, and PQ corresponds to AB. (1)

   Scale factor $=\frac{PQ}{AB}=\frac{9}{6}=1.5$. (1)

   QR corresponds to BC: $QR=8\times 1.5=12$. (1)

   PR corresponds to CA: $PR=10\times 1.5=15$. (1)

   Final answer: $QR=12$ and $PR=15$.

7. Question 7

   1 mark

   A model car is built at a scale of 1:20. If the model is 12 cm long, the real car is ___

   A. 240 cm

   B. 20 cm

   C. 32 cm

   D. 12 cm

   (1 mark)

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8. Question 8

   1 mark

   Two triangles are similar if their ___ are equal.

   A. Areas

   B. Angles

   C. Sides

   D. Perimeters

   (1 mark)

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9. Question 9

   3 marks

   Two similar cylinders have radii in the ratio 1:3. If the smaller cylinder has volume 20 cm³, find the volume of the larger cylinder.

   (3 marks)

   Marking scheme

   For similar solids, volumes are in the ratio of the linear scale factor cubed. (1)

   The radii are $1:3$, so the volumes are in the ratio $1^3:3^3=1:27$. (1)

   The larger volume is therefore $20\times 27=540$. (1)

   Final answer: $540$ cm${}^3$.

10. Question 10

    1 mark

    Two similar rectangles have lengths 5 cm and 15 cm. The shorter one has width 3 cm. The width of the longer one is ___

    A. 6 cm

    B. 45 cm

    C. 3 cm

    D. 9 cm

    (1 mark)

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11. Question 11

    1 mark

    Under an enlargement with scale factor −1 and centre origin, the image of (3, 4) is ___

    A. (4, 3)

    B. (−3, −4)

    C. (3, 4)

    D. (−4, −3)

    (1 mark)

    Marking scheme

    B. $(-3,-4)$ — a negative scale factor sends each point through the centre to the opposite side, so a factor of $-1$ turns the shape through $180^{\circ }$ about the origin. D and A swap the coordinates instead of changing their signs.

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