The radii of two concentric circles are 13 cm and 8 cm . AB is a diameter of the bigger circle BD is tangent to the smaller circle touching it at D.Find the length of AD
We draw a line parralel to BD from O on AD at M such that OM || BD
By thales' theorem or basic proportionality theroem , or similarity
AO/BO = AM/DM
But, AO = BO
=> AM/DM = 1 or AM = DM
Also, LMOD = LODM = 900 (alternate angles)
Now, BD2 = OB2 - OD2 = 169 - 64 =105
or, BD = √105 cm
Again, since OM || BD
so, by similarity
OM/BD = AO/AB
=> OM = BD X AO / AB = √105 x 13 / 26 = √105 / 2
or, OM = √105 / 2
So in triangle ODM
DM = √OM2 + OD2 = √105/4 + 64 = √105 + 256 / 2 = √361 / 2
or, DM = √361 / 2
Hence, AD = 2 DM = √361
hope this helps you frnd........