(ii) secant

(iii) two

(iv) point of contact

(B) 12 cm

(C) 15 cm

(D) 24.5 cm

(A) 60°

(B) 70°

(C) 80°

(D) 90°

(A) 50°

(B) 60°

(C) 70°

(D) 80°

[Using Pythagoras Theorem]

OP

In ΔOAP and ΔOAS,

AP = AS (Tangents from the same point)

OP = OS (Radii of the circle)

OA = OA (Common side)

ΔOAP ≅ ΔOAS (SSS congruence condition)

∴ ∠POA = ∠AOS

⇒∠1 = ∠8

Similarly we get,

∠2 = ∠3

∠4 = ∠5

∠6 = ∠7

Adding all these angles,

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 +∠8 = 360º

⇒ (∠1 + ∠8) + (∠2 + ∠3) + (∠4 + ∠5) + (∠6 + ∠7) = 360º

⇒ 2 ∠1 + 2 ∠2 + 2 ∠5 + 2 ∠6 = 360º

⇒ 2(∠1 + ∠2) + 2(∠5 + ∠6) = 360º

⇒ (∠1 + ∠2) + (∠5 + ∠6) = 180º

⇒ ∠AOB + ∠COD = 180º

Similarly, we can prove that ∠ BOC + ∠ DOA = 180º

Hence, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.