If (2, 7), (5, 1), (x, 3) are the vertices of a triangle whose area is 18 (sq. units), then find the possible value of x.

Correct option is C

Given:

Vertices of the triangle: ( A(2, 7) ), ( B(5, 1) ), ( C(x, 3) )

Area of the triangle = 18 square units.

Formula Used:

​$$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$​​

Solution:

18 = $$\frac{1}{2} \left| 2(1 - 3) + 5(3 - 7) + x(7 - 1) \right|$$​​

18 = $$\frac{1}{2} \left| 2(-2) + 5(-4) + x(6) \right|$$​

18 = $$\frac{1}{2} \left| -4 - 20 + 6x \right|$$​

18 = $$\frac{1}{2} \left| 6x - 24 \right|$$​​

36 = $$\left| 6x - 24 \right|$$​​

6x - 24 = 36 $$\quad \text{or} \quad 6x - 24 = -36$$​​

( 6x - 24 = 36)

6x = 60 $$\implies$$​ x = 10

( 6x - 24 = -36)

6x = -12 $$\implies$$​ x = -2

The possible values of ( x) are ( 10) and ( -2)