Question

A triangle on the coordinate plane has points located at a (2,5), b (5,9), and c (8,5). What is the area of the triangle. 

Answer 1

The area of the triangle is 12 unit²

Explanation:

The vertices of the triangle are;

a(2, 5), b(5, 9), and c(8, 5)

The area of a triangle given coordinates can be given by the following determinant;

`\Delta = (1)/(2)\begin{vmatrix}x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{vmatrix} = (1)/(2) \left | x_(1)\cdot y_(2) - x_(2)\cdot y_(1) + x_(2)\cdot y_(3) - x_(3)\cdot y_(2) + x_(3)\cdot y_(1) - x_(1)\cdot y_(3)\right |`

Therefore, we have;

`\Delta = (1)/(2)\begin{vmatrix}2 & 5 & 1\\ 5 & 9 & 1\\ 8 & 5 & 1\end{vmatrix} = (1)/(2) \left | 2* 9 - 5* 5 + 5* 5 - 8* 9 + 8* 5 - 2* 5 \right |`

Utilizing an online determinant calculator, we have;

`(1)/(2) * \left | -12 \right |= 12`

Therefore, the area of the triangle is 12 unit².