Answer:
Length of side of rhombus is [tex]x=\frac{ab}{a+b}[/tex]
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
[tex]\frac{EF}{AB}=\frac{CF}{AC}[/tex]
⇒ [tex]\frac{x}{a}=\frac{b-x}{b}[/tex]
⇒ [tex]xb=ab-ax[/tex]
⇒ [tex]x(a+b)=ab[/tex]
⇒ [tex]x=\frac{ab}{a+b}[/tex]
Hence, length of side of rhombus is [tex]x=\frac{ab}{a+b}[/tex]
