O-LevelMathematicsGeometry4024/21 · Oct/Nov 2023 · Q108 Marks

The diagram shows a circle, centre O, with diameters AC and BD.

(a)Show that triangle OAB is congruent to triangle ODC. Give a reason for each statement you make.[3]
(b)The diameter of the circle is 10cm and AB = 9cm. Calculate the difference between the circumference of the circle and the perimeter of the shaded shape.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[3] (a) AO = DO [equal] radii
[3] (a) BO = CO [equal] radii
[3] (a) ∠AOB = ∠DOC [vertically] opposite
[3] (a) [Congruent] SAS
[3] (a) B2 for two pairs of equal sides/angles with correct reasons or B1 for one pair of equal sides/angles with correct reason or for two appropriate pairs with no or incorrect reasons
[3] (a) Angles described using convention of 3 letters
[5] (b) 4.39 or 4.40 or 4.392 to 4.398
[5] (b) M2 for [AOB =]2xsin^-1(4.5/5) oe or cos[] = (5^2+5^2-9^2)/(2x5x5) oe or M1 for sin[] = 4.5/5 oe or 9^2 = 5^2 + 5^2 - 2x5x5cos[] oe
[5] (b) AND M2 for 10π - (9+9+2x(their AOD/360)x10π) oe or M1 for [2x](their AOD/360)x10π oe

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