Theory question 109
Circle Theorems(a) PQ is a tangent to a circle RST at the point S. PRT is a straight line, ∠TPS = 34° and ∠TSQ = 65°. (i) illustrate the information in a diagram. (ii) Find the value of: (A) ∠RTS; (B) ∠SRP
(b) In the diagram, |VZ| = |YZ|, ∠YXZ = 20° and ∠ZVY = 52°. Calculate the size of ∠WYZ.
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(a)(i) Circle through R, S, T; tangent PSQ at S; P outside the circle with P, R, T on one straight line (R between P and T).
(ii)(A) ∠PST = 180° − 65° = 115°; in triangle PST, ∠RTS = 180° − 34° − 115° = 31°
(B) ∠SRT = ∠TSQ = 65° (alternate segment), so ∠SRP = 180° − 65° = 115°
(b) ∠ZYV = ∠ZVY = 52° (|VZ| = |YZ|)
In triangle XYZ: ∠XZY = 180° − 20° − 52° = 108°
∠YWZ = ∠YVZ = 52° (same segment)
In triangle WYZ: ∠WYZ = 180° − 108° − 52° = 20°
Use the alternate segment theorem for the tangent at S.