Question

In figure, if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, then BD = ______.

Answer 1

In figure, if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, then BD = `underlinebb(55/3 cm)`.

Explanation:

Since (A, D, B) are collinear, (∠CDA = ∠CDB). 

Given (∠ACB = ∠CDA), we get ∠ACB = ∠CDB. 

So points (A, C, D, B) are concyclic equal angles subtend the same chord (AB).

Now use power of a point from (A) with secants (A–D–B) and (A–C): 

AD · AB = AC²

Given (AD = 3) and (AC = 8):

3 · AB = 64

⇒ `AB = 64/3`

Therefore,

BD = AB – AB

= `64/3 - 3`

= `55/3` cm