NCERT Example 3 — Circles

NCERT Example

Example 3

chord · tangent length

PQ is a chord of length 8 cm in a circle of radius 5 cm. The tangents at P and Q meet at T. Find TP.

Board-exam working:

Join OT and let it meet PQ at R.

Since TP = TQ, triangle TPQ is isosceles. OT bisects ∠PTQ and is perpendicular to PQ.

PR=RQ=\frac82=4\text{ cm}

In right triangle OPR:

OR=\sqrt{OP^2-PR^2}

=\sqrt{5^2-4^2}=3\text{ cm}

Triangles TRP and PRO are similar by AA similarity.

\frac{TP}{PO}=\frac{PR}{RO}

\frac{TP}{5}=\frac43

TP=\frac{20}{3}\text{ cm}

TP=\frac{20}{3}\text{ cm}