O-LevelMathematicsGeometry4024/22 · Oct/Nov 2024 · Q812 Marks

[Figure showing triangles ADB and AEC as straight lines. BC is parallel to DE. A diagram with two similar triangles ABC and ADE. BC = 9.8 cm, BD = 2.7 cm, DE = 5.6 cm and CE = 3.9 cm. Labelled 'NOT TO SCALE'.]

(a)ADB and AEC are straight lines. BC is parallel to DE. BC = 9.8 cm, BD = 2.7 cm, DE = 5.6 cm and CE = 3.9 cm. Complete the missing angles and reasons to show that triangle ABC is similar to triangle ADE. In triangle ABC and triangle ADE, angle BAC = angle __________ because common angle angle ABC = angle __________ because __________ angle ACB = angle __________ because __________ As the three pairs of angles are equal, triangle ABC is similar to triangle ADE.[3]
(b)Show that AD = 3.6 cm and AE = 5.2 cm.[4]
(c)Calculate angle DAE.[3]
(d)Calculate the area of triangle ABC.[2]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] for three correct angles in correct order (DAE, ADE, AED)
[B1] for line 2 correct (corresponding angles)
[B1] for line 3 correct (corresponding angles)
[4] Correct elimination of fraction and rearrangement leading to AD = 3.6 and AE = 5.2
[A2] with no errors seen leading to answers
[A1] for correct elimination of fraction and rearrangement leading to AD = 3.6 or AE = 5.2 with no errors seen leading to answer
[M2] for AD / (AD+2.7) = 5.6 / 9.8 or AE / (AE+3.9) = 5.6 / 9.8 oe
[M1] for 5.6 / 9.8 oe or 9.8 / 5.6 oe isw
[3] 76.7 or 76.65 to 76.66
[M2] for cos(DAE) = (3.6² + 5.2² - 5.6²) / (2 × 3.6 × 5.2) oe
[M1] for 5.6² = 3.6² + 5.2² - 2 × 3.6 × 5.2 × cos(DAE) oe
[2] 27.9 or 27.87 to 27.90
[M1] for 1/2 × (3.6 + 2.7) × (5.2 + 3.9) × sin(their76.7) oe
[SC1] If 0 scored, for answer 9.11 or 9.107 to 9.109

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