Theory question 112
Circle Theorems(a) In the diagram, PQRS is a circle. |PQ| = |QS|, ∠SPR = 26° and the interior angles of ΔPQS are in the ratio 2 : 3 : 3. Calculate: (i) ∠PQR; (ii) ∠RPQ; (iii) ∠PRQ.
(b) The coordinates of two points P and Q in a plane are (7, 3) and (5, x) respectively, where x is a real number. If |PQ| = √29 units, find the value of x.
Show the model answer
Model answer
(a) Angles of ΔPQS: 2/8 × 180° = 45° at Q; 67.5° at P and at S (equal sides PQ = QS)
(i) ∠SQR = ∠SPR = 26° (same segment); ∠PQR = 45° + 26° = 71°
(ii) ∠RPQ = ∠SPQ − ∠SPR = 67.5° − 26° = 41.5°
(iii) ∠PRQ = 180° − 71° − 41.5° = 67.5°
(b) (5 − 7)2 + (x − 3)2 = 29 → (x − 3)2 = 25
x − 3 = ±5 → x = 8 or x = −2
Explanation
The angle 45° is at Q, opposite the base PS of the isosceles triangle.