QQuestion
In the given diagram, O is the centre of the circle. Chord SR produced meets the tangent XTP at P.
(a) Prove that △PTR ∼ △PST.
(b) Prove that PT² = PR × PS.
(c) If PR = 4 cm and PS = 16 cm, find the length of tangent PT.
✓Answer
(a) By the alternate segment theorem:
∠PTR = ∠PST
Also, ∠RPT = ∠TPS, as these are the same angle at P.
Therefore, △PTR ∼ △PST by the AA criterion.
(b) Corresponding sides of similar triangles are proportional.
\frac{\mathrm{PT}}{\mathrm{PS}}=\frac{\mathrm{PR}}{\mathrm{PT}}⇒ PT² = PR × PS
Hence proved.
(c) PR = 4 cm and PS = 16 cm
PT² = PR × PS
⇒ PT² = 4 × 16
⇒ PT² = 64
⇒ PT = 8 cm
Therefore, the length of the tangent PT is 8 cm.
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