Q. 59 - Precalculus Enhanced with Graphing Utilities

Question

Geometry: Area of a Triangle A triangle has vertices $\left(x_1,y_1\right),\left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$. The area of the triangle is given by the absolute value of D, where

$D=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ 1& 1& 1\end{array}\right|$

Use this formula to find the area of a triangle with vertices $(2,3),(5,2)\text{, and }(6,5)$.

Step-by-Step Solution

Answer

The area of the triangle is $5$.

Step 1. Given Information

We are given three points $(2,3),(5,2)\text{, and }(6,5)$,

We have to find the area of triangle containing the given points.

Step 2. Finding the area of triangle

The area of triangle is given by,

$$\begin{gathered} D=\frac{1}{2}\left|\begin{array}{lll}2& 5& 6\\ 3& 2& 5\\ 1& 1& 1\end{array}\right| \\ =\frac{1}{2}[2·(2-5)-5·(3-5)+6·(3-2\left)\right] \\ =\frac{1}{2}[2·(-3)-5·(-2)+6·1] \\ =\frac{1}{2}(-6+10+6) \\ =5 \end{gathered}$$

Hence, the area of trianlge is $5$.