Question

$PQ$ is a chord of length $8cm$ of a circle of radius $5cm$ and center $O$. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length $TP$.

Answer 1

Join $OT$

In $\Delta OPT$

$O{T}^2=O{P}^2+P{T}^2\cdots \left(1\right)$

$OT$ is perpendicular bisector to $PQ$ let they met at $S$

$⟹SP=4$cm

By pythagorean theorm

$⟹SO=3$cm

By property associated with altitudes of Right angled triangle

$⟹\Delta OPT\approx \Delta PSO$

$⟹\frac{OS}{SP}=\frac{OP}{PT}⟹\frac{3}{4}=\frac{5}{TP}⟹TP=6.66$ cm