Answer:
The length of the rhombus is =[tex]\frac{bc}{b+c}cm[/tex]
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:
[tex]\frac{BE}{CE}=\frac{BA}{AC}=\frac{c}{b}[/tex]
⇒[tex]\frac{BE}{CE}=\frac{c}{b}[/tex]
⇒[tex]\frac{BE}{BC}=\frac{c}{c+b}[/tex]
Now, ΔDBE is similar to ΔABC, then
DE=[tex](\frac{BE}{BC}){\times}AC[/tex]
=[tex](\frac{c}{c+b}){\times}b[/tex]
=[tex]\frac{bc}{b+c}cm[/tex]
Thus, the length of the rhombus is =[tex]\frac{bc}{b+c}cm[/tex]
