Question

A circle is inscribed in a right-angled triangle, as shown. The point of contact of the circle and the hypotenuse divides the hypotenuse into lengths x and y. Prove that the area of the triangle is equal to xy.

• A. undefined
• B. undefined
• C. undefined
• D. undefined

Answer 1

Applying pythagoras theorem:
$(x+r{)}^2+(y+r{)}^2=(x+y{)}^2\Rightarrow (x^2+2xr+r^2)+(y^2+2yr+r^2)=x^2+2xy+y^{2}2r(x+y)+2r^2=2xyr(x+y)+r^2=xy.....(1)$
Area of triangle $=\frac{1}{2}(r+y)(x+r)(\frac{1}{2}\times base\times hight)$
$=\frac{1}{2}\left[r\right(r+y)+r^2+xy]\left[From\right(1\left)\right]=\frac{1}{2}[xy+xy]=\frac{2xy}{2}=xy$
Hence Proved.