The two sides of a triangle measure 10 cm and 8 cm respectively. The area of the triangle is equal to $$16\sqrt6$$​ $$\text{cm}^2$$​. Find the length of the third side, given that it is a rational number.

Correct option is C

Given:

Two sides of the triangle are 10 cm and 8 cm.

The area of the triangle is 16$$\sqrt 6$$​ cm².

The length of the third side is a rational number.

We need to find the length of the third side.

Formula Used:

The area of a triangle can be calculated using Heron's formula:

A = $$\sqrt{s(s-a)(s-b)(s-c)}$$​

where:

a, b, and c are the sides of the triangle.

s is the semi-perimeter, given by:

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

Solution:

s = $$\frac {18+2x}2$$ = 9 + x

A =$$\sqrt{(9+x)(9+x-10)(9+x -8)(9+x-2x)}$$ = $$16\sqrt 6$$​​

​$$\sqrt{(9+x)(x-1)(x +1)(9-x)}$$ = $$16\sqrt 6$$

​$$\sqrt{(x^2-1)(81-x^2)}$$ = $$16\sqrt 6$$​​

Squaring both sides,

(x² - 1) (81 - x²​) = 1536

By putting 2x = 14 then, x = 7

L.H.S.

(49 - 1)(81 - 49)

= 48 $$\times$$ 32 = 1536    R.H.S.

Only Option (c) satisfied.

Thus, option (c) is right.