Question

The sides of a triangle (in cm) are x, x+1, 2x-1 and its area is xroot10 . Find the value of x and the lengths of the sides of the triangle

Answer 1

Answer

The sides of triangle are 6, 7 and 11 cm

Explanation

The sides of triangle are x, (x + 1) and (2x - 1)

⇒ perimeter of triangle = x + (x + 1) + (2x - 1)

⇒ 4x

⇒ semi perimeter of triangle (s) = 4x/2 = 2x

Now, by heron's formula, we have the area as

$\sf\sqrt{s(s-a)(s-b)(s-c)}$

This gives the area of triangle as

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

But, the area given is x√10, so the two quantities should be equal

$\sf x\sqrt{2(x-1)}=x\sqrt{10}\\\\\implies \sqrt{2(x-1)}=\sqrt{10}\\\\\implies 2(x - 1) = 10\\\\\implies 2x - 2 = 10\\\\\implies 2x = 12 \\\\\implies x = 6$

So, the sides are x, (x + 1) and (2x - 1)

⇒ 6, (6 + 1) and (12 - 1)

⇒ 6, 7, 11