Question

the radii of two concentric circles are 13cm and 8cm AB is a diameter of the bigger circle BD is a tangent to the smaller circle touching at D.Find the length of AD.

Answer 1

Refer the attachment.
From the right triangle ODB,
$cos\theta = \frac{8}{13}$
For the triangle AOD, applying cosine rule,
AD^2 = AO^2 + OD^2 - 2×AO×OD×cos(AOD)

$= {13}^{2} + {8}^{2} -2 \times 13 \times 8 \times \cos(\pi - \theta )$
$= 169 + 64 - 2 \times 13 \times 8 \times ( - \cos( \theta ) )$
$= 169 + 64 + 2 \times 13 \times 8 \times \frac{8}{13} = 361$
Thus AD = √361 = 19 cm