Question

The sides of a triangle are x,x+1,2x-1 and it’s area is xroot 10. Find the value of x

Answer 1

Required Answer:-

Given Information:

• The sides of a triangle are x, x + 1 and 2x + 1
• Area of the triangle is x√10 sq unit.

To find:

• The value of x.

Answer:

• The value of x is 6 unit.

Solution:

We shall use Heron's Formula to solve the given question.

Heron's Formula:-

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

Where,

1. A = Area of triangle.
2. s = Semi-Perimeter of the triangle.
3. a, b and c are the sides of the triangle.

Here,

➡ a = x

➡ b = x + 1

➡ c = 2x - 1

So, Semi-Perimeter

= (a + b + c)/2

= (x + x + 1 + 2x - 1)/2

= 4x/2

= 2x

Now,

Area of the triangle = x√10

Therefore, according to the given conditions,

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

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

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

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

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

Squaring both sides, we get,

$\sf \implies 10= 2(x- 1)$

Now, let's solve the linear equation,

$\sf \implies 5= (x- 1)$

$\sf \implies x = 5 + 1$

$\sf \implies x = 6$

Hence, the value of x is 6 unit.