Question

In the given diagram ‘O’ is the centre of the circle. Chord SR produced meets the tangent XTP at P.

1. Prove that ΔPTR ~ ΔPST
2. Prove that PT² = PR × PS
3. If PR = 4 cm and PS = 16 cm, find the length of the tangent PT.

Answer 1

(a) Since PT is tangent the circle at point T, We know that,

∠PST = ∠PTR (Angles in the alternate segment)

∠P = ∠P   .....(common)

Therefore, by AA critertion,

ΔPTR ~ ΔPST  Hence Proved.

(b) Since, ΔPTR ~ ΔPST

So, cooresponding sides are in proportion,

`(PT)/(PS) = (PR)/(PT)`

PT² = PR × PS Hence Proved.

(c) Given, PR = 4 cm, PS = 16 cm

Now, 

PT² = PR × PS

PT² = 4 × 16

= 64

PT = 8 cm