Question

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

Answer 1

Given,

AC = 8 cm,

AD = 3 cm

And ∠ACB = ∠CDA

From figure,

∠CDA = 90°

∠ACB = ∠CDA = 90°

In right angled ΔADC, 

AC² = AD² + CD² 

⇒ (8)² = (3)² + (CD)²

⇒ 64 – 9 = CD²

⇒ CD = `sqrt(55)` cm

In ΔCDB and ΔADC,

∠BDC = ∠ADC  ...[Each 90°]

∠DBC = ∠DCA   ...[Each equal to 90° – ∠A]

∴ ΔCDB ∼ ΔADC

Then, `("CD")/("BD") = ("AD")/("CD")`

⇒ CD² = AD × BD

∴ BD = `("CD"^2)/("AD")`

= `(sqrt(155))^2/3`

= `55/3` cm