In the adjoining figure, CBA is a secant and CD is tangent to the circle. If AB = 7 cm and BC = 9 cm, then

(i) Prove that △ACD ~ △DCB

(ii) find the length of CD.

(i) In △ACD and △DCB

∠C = ∠C (Common angles)

∠CAD = ∠CDB (Angles in alternate segments are equal)

∴ △ACD ~ △DCB (By AA axiom.)

Hence, proved that △ACD ~ △DCB.

(ii) Since triangles are similar hence, the ratio of their corresponding sides are equal.

$\dfrac{AC}{DC} = \dfrac{DC}{BC} \\[1em] DC^2 = AC \times BC \\[1em] DC^2 = (AB + BC) \times BC \\[1em] DC^2 = (7 + 9) \times 9 \\[1em] DC^2 = 16 \times 9 \\[1em] DC^2 = 144 \\[1em] DC = \sqrt{144} \text{ cm} \\[1em] DC = 12 \text{ cm}.$

Hence, the length of DC = 12 cm.