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

Length of side of rhombus is
`x=(ab)/(a+b)`

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

`(EF)/(AB)=(CF)/(AC)`

⇒
`(x)/(a)=(b-x)/(b)`

⇒
`xb=ab-ax`

⇒
`x(a+b)=ab`

⇒
`x=(ab)/(a+b)`

Hence, length of side of rhombus is
`x=(ab)/(a+b)`