WAEC Mathematics 1998 Theory question 111

Theory question 111

Mensuration of Plane Shapes

In the diagram, ASRTB represents a piece of string passing over a pulley of radius 10cm in a vertical plane. O is the centre of the pulley and AMB is a horizontal straight line touching the pulley at M. Angle SAB = 90° and angle TBA = 60°.
(a) Calculate (i) the obtuse angle SOT, (ii) arc SRT (iii) |BT|.
(b) Find, correct to the nearest cm, the length of the string. [Take π = 22/7].

Show the model answer

Model answer

(a)(i) ASOM is a square (radii meet tangents at 90°), so ∠SOM = 90°. In OMBT, ∠MOT = 360 − 90 − 90 − 60 = 120°. Obtuse ∠SOT = 360 − 90 − 120 = 150°.
(ii) Arc SRT = 150/360 × 2 × 22/7 × 10 ≈ 26.19 cm.
(iii) OB bisects ∠TBM, so ∠OBT = 30° and BT = 10/tan 30° = 10√3 ≈ 17.32 cm.
(b) String = AS + arc SRT + TB = 10 + 26.19 + 17.32 = 53.51 ≈ 54 cm.

Explanation

Tangents from a point are equal, and a radius meets a tangent at 90°.