Question

If PA and PB are tangents from an outside point P. such that PA =10 cm and $\angle APB=60$. Find the length of chord AB.

Answer 1

Consider a circle C with center O.

We have PA and PB are tangents of the circle, PA=10cm and

$\angle APB={60}^o$

Join OP,

Such that,

In $△PAC$ and $△PBC$ we have,

$PA=PB$ [tangent of the circle fro the outer point p is equal]

$\angle PAC=\angle PBC$ [angle made by the external tangent on a

circle is equal]

$PC=CP$ [common]

so,

$△PAC$ $\cong$ $△PBC$ [By SAS criteria]

so,

$AC=BC$ ........(i)

$\angle ACP=\angle BCP$ ......(ii)

since

$\angle APB=\angle APC+\angle BPC$

so,

$\angle APC=\frac{1}{2}\times {60}^0={30}^0$

$\angle APC={30}^0$

$\angle ACP+\angle BCP={180}^0$

from equation 2 we get

$\angle ACP=\frac{1}{2}\times {180}^0={90}^0$

Thus in Right $△$ ACP

$sin$${30}^0=\frac{AC}{AP}$

$\frac{1}{2}=\frac{AC}{10}$cm

$AC=5$cm

Since

$AC=BC$

so,

$AB=AC+BC$

$=5$cm$+5$cm

$=10$cm