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0580/31 May/Jun 2021, Question 6: Describing Transformations

Cambridge IGCSE Mathematics 0580 · Paper 31 (Core) · Calculator · 14 marks

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The question, in short

Using a 1 cm² grid with triangles A, B and C, describe transformations, draw images, use trigonometry to show angle b is 63.4°, and decide whether triangles B, D and E are all similar.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 3 marks

Describe fully the single transformation mapping triangle A onto triangle B.

  1. 1B is bigger than A, so it is an enlargement.
  2. 2A has base 1 and B has base 4, so scale factor = 4
  3. 3Lines joining matching vertices, e.g. (−2, −3) to (4, 3), meet at (−4, −5)
  4. 4Enlargement, centre (−4, −5), scale factor 4

AnswerEnlargement, centre (−4, −5), scale factor 4

Part (aii) · 3 marks

Describe fully the single transformation mapping triangle A onto triangle C.

  1. 1Rotating each vertex of A 90° clockwise about (0, 0): (x, y) → (y, −x)
  2. 2(−3, −5) → (−5, 3), (−2, −5) → (−5, 2), (−2, −3) → (−3, 2)
  3. 3Rotation, centre (0, 0), 90° clockwise

AnswerRotation, centre (0, 0), 90° clockwise

Part (bi) · 2 marks

Draw triangle A's image after translation by column vector (−5, 4).

  1. 1Subtract 5 from each x-coordinate and add 4 to each y-coordinate.
  2. 2(−3, −5) → (−8, −1), (−2, −5) → (−7, −1), (−2, −3) → (−7, 1)
  3. 3Draw the triangle with vertices (−7, 1), (−7, −1), (−8, −1)

AnswerTriangle at (−7, 1), (−7, −1), (−8, −1)

Part (bii) · 2 marks

Draw triangle A's image after reflection in the line x = −4.5.

  1. 1Each point moves to the same distance on the other side of x = −4.5: new x = −9 − x
  2. 2(−3, −5) → (−6, −5), (−2, −5) → (−7, −5), (−2, −3) → (−7, −3)
  3. 3Draw the triangle with vertices (−6, −5), (−7, −3), (−7, −5)

AnswerTriangle at (−6, −5), (−7, −3), (−7, −5)

Part (c) · 2 marks

Use trigonometry to show that angle b in triangle B is 63.4°, correct to 1 decimal place.

  1. 1Opposite side = 3 − (−5) = 8, adjacent side = 4 − 0 = 4
  2. 2tan b = 8/4 = 2
  3. 3b = tan⁻¹(2) = 63.43… = 63.4° to 1 decimal place

Answertan b = 8/4 so b = 63.43…

Part (d) · 2 marks

Decide whether all three triangles B, D and E are similar, giving a reason.

  1. 1Triangle B: angles 90°, 63.4° and 180 − 90 − 63.4 = 26.6°
  2. 2Triangle E: 90° and 63.4°, so its third angle is 26.6°
  3. 3Triangle D: 90° and the 26.6° angle it shares with B, so its third angle is 63.4°
  4. 4Yes, all three triangles have angles 90°, 63.4° and 26.6°, so they are similar

AnswerYes, all three triangles have angles 90°, 63.4° and 26.6°

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