O-LevelMathematicsGeometry4024/21 · Oct/Nov 2018 · Q69 Marks
A, B, C and D are points on the circle centre O. TA and TC are tangents to the circle. TDOB is a straight line. ATO = x°. [Figure 6.1]
- (a)Show that triangle OAT is congruent to triangle OCT. Give a reason for each statement you make.[3]
- (b)(i)Find, in terms of x, giving your answers as simply as possible AỘT,[1]
- (b)(ii)Find, in terms of x, giving your answers as simply as possible OÂB,[1]
- (b)(iii)Find, in terms of x, giving your answers as simply as possible reflex AĎC.[1]
- (c)The radius of the circle is 6 cm. Given that x = 35°, work out BT.[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 2018). The mark or mark type is shown in brackets.
- [3] ∠TAO= ∠TCO tangent perpendicular to radius
- AO = CO [equal] radii
- TO is common
- Congruent RHS
- B1 for one correct pair of equal angles and one correct pair of equal sides
- or for two correct pairs of equal sides
- B1 for correct reason for two pairs of equal sides/angles
- [1] 90 - x
- [1] (90 - x) / 2 oe
- FT their algebraic (b)(ii) ÷ 2
- [1] 270 - x
- [3] 16.5 or 16.46[...]
- M2 for [OT = ] 6 / sin 35
- or M1 for sin 35 = 6 / [OT]
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