O-LevelMathematicsGeometry4024/22 · Oct/Nov 2020 · Q710 Marks

(a) [Figure 7.1] The diagram shows part of an n-sided regular polygon ABCDEFGH... DCF = EFC = 24°.

(a)(i)Find the value of n.[2]
(a)(ii)Find HFG.[2]
(b)(i)Show that triangle PQS is similar to triangle TQX. Give a reason for each statement you make.[3]
(b)(ii)Find the ratio SX: SQ.[1]
(b)(iii)Find the ratio area of triangle TQX: area of parallelogram PQRS.[2]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[2] B1 for exterior angle = 24° or interior angle = 156°; or M1 for 360/24 soi or 180(n-2)/n = 180-24 oe; Final answer n=15.
[2] M1 for x + x + 156 = 180 oe; Final answer 12°.
[3] B2 for two correct pairs of angles with correct reasons; or B1 for one pair with a correct reason or two correct pairs of angles identified with wrong/no reasons; ∠Q is common / shared / same (AA similarity); ∠PSQ = ∠TXQ (corresponding angles, TU || QR); ∠SPQ = ∠XTQ (corresponding angles, TU || QR); Hence, triangle PQS is similar to triangle TQX.
[1] 2:3.
[2] B1 for [1/2 * (1/3) * (1/3)] : 1 oe (considering ratio of areas of triangles with common height or using ratio of sides squared for similar triangles); After 0 scored SC1 for (1/3)^2 : 1; Final answer 1:18.

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