Question

In the given figure, AB = 7 cm and BC = 9 cm.

1. Prove that: AACD −  ADCB
2. Find the length of CD.

Answer 1

Given

AB = 7 cm

BC = 9 cm

A, B, and D lie on the circle, and C is an external point.

(i) To prove: ∠ACD = ∠ADB

From external point C, the two secants C–A–B and C–D–B cut the circle at A, B and D, B.

For such secants, the exterior angle formed at C equals the angle inside the circle standing on the same arc.

Here, both ∠ACD and ∠ADB stand on arc AD.

∠ACD = ∠ADB.

(ii) To find CD

AC = AB + BC

AC = 7 + 9 = 16 cm

Using the Secant–Secant Theorem:

BC × AC = CD²

9 × 16 = CD²

144 = CD²

CD = 12 cm