Question

In ∆ABC, XY//AC and XY divides the triangle into two parts of equal area find the ratio of AX/XB

Answer 1

Solution:

In  ∆ABC, XY║BC and XY divides the triangle into two parts of equal area.

So, Area (ΔABC)=2 area (ΔAXY)------(1)

In Δ AXY and ΔABC

∠A is common.

∠AXY = ∠ABC→→XY║BC, And these are corresponding angles of two triangles.

So, ΔAXY ~ ΔABC→→→[Angle angle similarity]

So, when triangles are similar their corresponding sides are proportional.

and square of corresponding sides is equal to ratio of areas of two triangles.

$\frac{Area{\Delta AXY}}{Area\Delta ABC}}=[\frac{AX}{XB}]^2=[\frac{AY}{YC}]^2\\\\ \frac{1}{2}=[\frac{AX}{XB}]^2\\\\ \frac{AX}{XB}=\frac{1}{\sqrt{2}}$

------------------------------using (1)