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:

The length of the rhombus is =$\frac{bc}{b+c}cm$

Step-by-step explanation:

It is given that the Rhombus ADEF is inscribed into a triangle ABC so that they share angle A and the vertex E lies on the side BC.Then,

AE is the angle bisector of ∠A, so divides the sides of the triangle into a proportion:

$\frac{BE}{CE}=\frac{BA}{AC}=\frac{c}{b}$

⇒$\frac{BE}{CE}=\frac{c}{b}$

⇒$\frac{BE}{BC}=\frac{c}{c+b}$

Now, ΔDBE is similar to ΔABC, then

DE=$(\frac{BE}{BC}){\times}AC$

=$(\frac{c}{c+b}){\times}b$

=$\frac{bc}{b+c}cm$

Thus, the length of the rhombus is =$\frac{bc}{b+c}cm$