In figure, 𝑃𝑄 is a chord of length 16 𝑐𝑚, of a circle of radius 10 𝑐𝑚. The tangents at 𝑃 and 𝑄 intersect at a point 𝑇. Find the length of 𝑇𝑃.

Given that,

 𝑃𝑄 is a chord of length 16 𝑐𝑚 

Also, radius  10 𝑐𝑚 

The tangents at 𝑃 and 𝑄 intersect at a point 𝑇. 

We have to find the length of 𝑇𝑃.

 Here, 𝑂𝑃 = 10 𝑐𝑚 

𝑃𝑀 = 𝑀Q

$=\frac{8}{2}$

= 4 𝑐𝑚 

In ∆𝑂𝑀𝑃,

By using Pythagoras Theorem 

𝑂𝑃² = 𝑃𝑀² + 𝑂𝑀² 

10² = 8² + 𝑂𝑀² 

𝑂𝑀² = 100 − 64 

𝑂𝑀² = 36 

𝑂𝑀 = 6 𝑐𝑚 

Let ∠𝑃𝑂𝑀 be θ

$\begin{array}{l}\tan \theta =\frac{PM}{MO}\\ =\frac{8}{6}\end{array}$

In ∆𝑂𝑇𝑃,

$$\begin{gathered} 𝑐𝑜𝑡 \theta = \frac{𝑂𝑃}{ 𝑇𝑃} \\ = \frac{10 }{ 𝑇𝑃} \end{gathered}$$

We know that $𝑂𝑃\perp 𝑇𝑃$ 

Radius is always perpendicular to Tangent.

$Now, we know that: \cot \theta =\frac{1}{\tan \theta }$

$\Rightarrow 𝑐𝑜𝑡 \theta =\frac{ 6 }{8} (\because 𝑡𝑎𝑛 \theta = 8 6 )$

$\Rightarrow \frac{ 6 }{8} = \frac{10}{ 𝑇𝑃}$

$\Rightarrow 𝑇𝑃 = \frac{80 }{6}$

$\Rightarrow 𝑇𝑃 = 13. 33 𝑐𝑚$

$Hence, the length of 𝑇𝑃 is 13.33 cm$