Question

Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC . What is the length of the side of the rhombus if AB=c, and AC=b.

Answer 1

Answer:

Length of side of rhombus is $x=\frac{ab}{a+b}$  

Step-by-step explanation:

Given Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC. We have to find the length of side of rhombus.

It is also given that AB=a and AC=b

Let side of rhombus is x.

In ΔCEF and ΔCBA

∠CEF=∠CBA        (∵Corresponding angles)

∠CFE=∠CAB        (∵Corresponding angles)

By AA similarity rule, ΔCEF~ΔCBA

∴ their sides are in proportion

$\frac{EF}{AB}=\frac{CF}{AC}$

⇒ $\frac{x}{a}=\frac{b-x}{b}$

⇒ $xb=ab-ax$

⇒ $x(a+b)=ab$

⇒ $x=\frac{ab}{a+b}$

Hence, length of side of rhombus is $x=\frac{ab}{a+b}$