Question

The sides of a triangle are $x,x+1,2x-1$ and its area is $x\sqrt{10}$. What is the value of $x$?

Answer 1

If $a,b,c$ are the sides of a triangle , then according to Hero's Formula area of triangle is $\sqrt{s(s-a)(s-b)(s-c)}$

where $s=\frac{a+b+c}{2}$

For the given triangle,

$s=\frac{x+x+1+2x-1}{2}$

$=\frac{4x}{2}=2x$

Area $=x\sqrt{10}$

$\sqrt{2x(2x-x)(2x-x-1)(2x-2x+1)}=x\sqrt{10}$

$\sqrt{2x\left(x\right)(x-1)\left(1\right)}=x\sqrt{10}$

$x\sqrt{2(x-1)}=x\sqrt{10}$

$\sqrt{2(x-1)}=\sqrt{10}$

$\Rightarrow 2(x-1)=10$

$x-1=5$

$x=6$