Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :

∠PAQ = 2∠OPQ

ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with center O, has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.

In the given figure, PT touches the circle with center O at point R. Diameter SQ is produced to meet the tangent TR at P.

Given ∠SPR = x° and ∠QRP = y°;

prove that :

(i) ∠ORS = y°

(ii) write an expression connecting x and y.

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate :

(i) ∠CBT

(ii) ∠BAT

(iii) ∠APT