Question

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

Answer 1

Given, AC = 8 cm, AD = 3cm

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 = CD^2`

⇒ `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)`

⇒ `CD^2 = AD xx BD`

∴ `BD = (CD^2)/(AD) = (sqrt(155))^2/3 = 55/3` cm