Question

ab is a chord of length 16cm of a circle of radius 10 cm .the tangents at a and b intersect at point p.find the length of pa





....Please solve it by Pythagoras theorem

Answer 1

The length of AP is 13.34 cm.

Given:

Length of the chord $AB=16cm$

Radius of the circle $=10cm$

To find:

Length of $AP$.

Solution:

We have been given a circle with center $O$ having a radius $R=10cm$ and a chord of length $AB=16cm$.

From the diagram, we can see that on joining $O$ to $P$, the line $OP$ intersects the chord $AB$ at $D$ which will be the perpendicular bisector of $AB$. Hence the length of $AD=BD=\frac{AB}{2}=8cm$.

In Δ$AOD$, applying Pythagoras theorem, we get

$OA^{2}= OD^{2}+ DA^{2}$

$R^{2}= OD^{2}+ AD^{2}$

$(10)^{2}= OD^{2}+ (8)^{2}$

$OD^{2}=100-64=36$

$OD=6cm$

Now,

We know, a tangent makes an angle of $90^{o}$ with the radius of the circle at the point of contact. In Δ$AOD$, let ∠$AOD=\alpha$

$sin\alpha =\frac{AD}{OA}$

$sin\alpha =\frac{8}{10}$

$sin\alpha =\frac{4}{5}$  

Hence, $tan\alpha =\frac{4}{3}$

Similarly, in Δ$AOP$, we have

$tan\alpha =\frac{AP}{OA}$

Equating both the values of $tan\alpha$, we get

$\frac{AP}{OA}= \frac{4}{3}$

$\frac{AP}{10}= \frac{4}{3}$

$AP=\frac{40}{3}$

$AP=13.34cm$

Final answer:

Hence, the length of AP is 13.4 cm.