Question

If the sides of a triangle are x,(x+1),(2x-1) and its area is x$\sqrt{10}$ sq units. then x = ______ units.

• A. 6

Answer 1

The correct option is A 6
Since the sides of the triangle are x, (x+1) and (2x-1);
$\text{Semi-perimeter},s=\frac{x+(x+1)+(2x-1)}{2}$
$s=2x$

Area of the triangle is given by Heron's formula:
$\sqrt{s\times (s-a)\times (s-b)\times (s-c)}$

Given, area of the triangle = x$\sqrt{10}$ sq. units.
$⟹x\sqrt{10}=\sqrt{s\times (s-a)\times (s-b)\times (s-c)}$
$⟹x\sqrt{10}=\sqrt{2x\times (2x-x)\times (2x-(x+1\left)\right)\times (2x-(2x-1\left)\right)}$
$⟹x\sqrt{10}=\sqrt{2x\times \left(x\right)\times (x-1)\times \left(1\right)}$

On squaring both the sides, we get,
$x^2\times 10=2x\times x\times (x-1)$
$⟹10x^2=2x^2\times (x-1)$
$⟹5=x-1$
$⟹x=6\text{units}$