Question

In ∆ABC, B - D - C and BD = 7, BC = 20 then find following ratio. 

${\displaystyle \frac{\text{A(∆ ABD)}}{\text{A(∆ ADC)}}}$

Answer 1

Construction: Draw a perpendicular from vertex A to line BC. 

BC = BD + DC   ...[B - D - C]

DC = BC − BD

DC = 20 − 7

DC = 13

Ratio of areas of two triangles is equal to the ratio of the products of their bases and corresponding heights.

∴ ${\displaystyle \frac{\text{A(∆ ABD)}}{\text{A(∆ ADC)}}=\frac{\frac{1}{2}\times \text{AX}\times \text{BD}}{\frac{1}{2}\times \text{AX}\times \text{DC}}}$

∴ ${\displaystyle \frac{\text{A(∆ ABD)}}{\text{A(∆ ADC)}}=\frac{\text{BD}}{\text{DC}}}$

∴ ${\displaystyle \frac{\text{A(∆ ABD)}}{\text{A(∆ ADC)}}=\frac{7}{13}}$