Question

The diagram on the right shows a circle.given that the length of arcs RS=2QR , angle QPR=35° and anglePSQ=45°,determine the value of

(a) angle SPR
(b) angle SRP

Answer 1

9514 1404 393

Answer:

(a) ∠SPR = 70°

(b) ∠SRP = 30°

Explanation:

The applicable relationships are ...

• the measure of an arc is twice the measure of the inscribed angle it subtends
• the sum of measures of the arcs of a circle is 360°
• the sum of measures of the angles of a triangle is 180°

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QR is subtended by 35° inscribed angle QPR, so is 2×35° = 70°.

PQ is subtended by 45° inscribed angle PSQ, so is 2×45° = 90°.

RS is 2×QR, so is 2×70° = 140°.

(a) Inscribed angle SPR is half the measure of arc RS, so is 140°/2 = 70°.

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(b) Arc SP is the remaining arc in the circle, so is ...

arc SP = 360° -arc PQ -arc QR -arc RS = 360° -90° -70° -140° = 60°.

Inscribed angle SRP is half the measure of arc SP, so is 60°/2 = 30°