Question

O is the centre of a circle of radius 6 cm. P is a
point such that OP = 10 cm and OP intersects the
circle at T and PC, PD are two tangents drawn to
the circle. If AB is the tangent to the circle at T,
find the length AB.

Answer 1

Given : O is the centre of a circle of radius 6 cm. P is a  point such that OP = 10 cm and OP intersects the  circle at T and PC, PD are two tangents drawn to  the circle. If AB is the tangent to the circle at T

To Find : the length AB​

Solution:

OC = OD = OT = 6 cm = Radius

OP = 10 cm

PT = OP - OT = 10 - 6  = 4 cm

PC is Tangent

Hence  OP² = OC² + PC²

=> 10² = 6² + PC²

=> 100 = 36 + PC²

=> 64 = PC²

=> PC = 8

similarly PD = 8   ( PC = PD Equal tangents )

Let say  AT  = x   cm

Then  AC = AT  = x    ( Equal Tangent )

AP = PC  - AC

=> AP = 8  - x

AT = c

∠ATP = 90° as ∠OTA = 90°  

AP² = AT² + PT²

=> (8 - x)² = x²  +  4²

=>  8² + x² - 16x = x² + 16

=> 64 - 16x = 16

=> 4 - x = 1

=> x = 3

AT = 3 cm

Similarly BT = 3 cm

AB = AT + BT  = 3 + 3 = 6 cm

AB = 6 cm

length AB​ = 6 cm

Hope this helps <3