Theory question 108
Mensuration of Plane Shapes(a) In the diagram, XY is a chord of a circle of a radius 5 cm. The chord subtends an angle 96° at the centre. Calculate, correct to three significant figures, the area of the minor segment cut-off by XY, (Take π = 22/7)
(b) The figure shows a circle of radius 7 cm inscribed in a square. If a portion of the circle is shaded with some portions of the square, calculate the total area of the shaded portions. (Take π = 22/7)
(c) In the diagram P, Q, R and S are points on the circumference of the circle centre O. ∠POQ = 160°, ∠QSR = 45° and ∠PQS = 40°. Calculate (i) ∠QPS, (ii) ∠RQS
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(a) Sector = 96/360 × 22/7 × 52 = 20.95 cm2. Triangle = ½ × 5 × 5 × sin 96° = 12.43 cm2. Segment = 20.95 − 12.43 ≈ 8.52 cm2.
(b) Square = 14 × 14 = 196 cm2; circle = 22/7 × 49 = 154 cm2; four corners = 42 cm2.
80° sector = 80/360 × 154 = 34.2 cm2. Total shaded = 42 + 34.2 ≈ 76.2 cm2.
(c)(i) ∠PSQ = ½ × 160° = 80°, so ∠QPS = 180 − 80 − 40 = 60°.
(ii) ∠QPR = ∠QSR = 45° (same arc QR), so ∠RPS = 60 − 45 = 15°, and ∠RQS = ∠RPS = 15° (same arc RS).
Minor segment = sector − triangle.