Theory question 111
Circle Theorems(a) PQRST is a circle with centre C. PCS is a straight line, RS//QT, |QR| = |RS| and ∠QTS = 56°. Find (i) ∠SQT (ii) ∠PQT
(b) In the diagram, points B and C are on a horizontal plane and |BC| = 30cm. A and D are points vertically above B and C respectively. |DC| = 40cm and |AB| = 26cm. Calculate the angles of depression of: (i) B from D; (ii) A from D: correct to the nearest degree.
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(a)(i) ∠QPS = ∠QTS = 56° (same segment). PQRS is cyclic, so ∠QRS = 180 − 56 = 124°.
Triangle QRS is isosceles (QR = RS), so ∠RSQ = (180 − 124)/2 = 28°.
RS ∥ QT, so ∠SQT = ∠RSQ = 28° (alternate angles).
(ii) PS is a diameter, so ∠PQS = 90°. ∠PQT = 90 − 28 = 62°.
(b)(i) The angle of depression of B from D equals ∠DBC: tan = 40/30, angle ≈ 53.1° ≈ 53°.
(ii) A is 40 − 26 = 14 cm below D over a horizontal 30 cm: tan = 14/30, angle ≈ 25.0° ≈ 25°.
An angle of depression is measured down from the horizontal at the observer.