O-LevelMathematicsGeometry4024/21 · Oct/Nov 2017 · Q1112 Marks

ABCD is a rectangle. P and Q are points on DC. AQ and BP intersect at R. Prove that triangle ARB is similar to triangle QRP. Give a reason for each statement you make.

(a)ABCD is a rectangle. P and Q are points on DC. AQ and BP intersect at R. Prove that triangle ARB is similar to triangle QRP. Give a reason for each statement you make.[3]
(b)(i)Calculate AQ.[2]
(b)(ii)Calculate AR.[2]
(b)(iii)Calculate the area of triangle ARB.[2]
(b)(iv)Calculate the total area shaded in the rectangle.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[3] B1 for one pair of angles stated with reason or for two pairs with no reasons or incorrect reasons
[3] B1 for a further correct pair of angles with reason
[3] ∠ARB = ∠PRQ, [vertically] opposite ∠RAB = ∠RQP, alternate [angles] ∠RBA = ∠RPQ alternate [angles] ΔARB and ΔQRP similar, equal angles
[2] M1 for cos 55 = 5/AQ or sin 35 = 5/AQ oe
[2] [AQ = ] 8.72 or 8.717[...]
[2] M1 for cos 35 = AR/9 or sin 55 = AR/9 oe
[2] [AR = ] 7.37[2...]
[2] M1 for 1/2 × their 7.37 × 9 × sin 35 oe
[2] Or 1/2 × their 7.37 × √9²-(their 7.37)²
[2] [Area ARB = ] 18.8 to 19.2[...]
[3] M1 for their area ARB + 1/2 × their RQ × their PR
[3] 19.6 to 19.7 nfww

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.