Question

given △def with coordinates d(2,3), e(4,7), and f(8,1). prove that △def is a right triangle

Explanation:

Step1: Find the slopes of the sides

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
For side $DE$ with $D(2,3)$ and $E(4,7)$:
$m_{DE}=\frac{7 - 3}{4 - 2}=\frac{4}{2}=2$.
For side $EF$ with $E(4,7)$ and $F(8,1)$:
$m_{EF}=\frac{1 - 7}{8 - 4}=\frac{-6}{4}=-\frac{3}{2}$.
For side $DF$ with $D(2,3)$ and $F(8,1)$:
$m_{DF}=\frac{1 - 3}{8 - 2}=\frac{-2}{6}=-\frac{1}{3}$.

Step2: Check for perpendicular - sides

Two lines are perpendicular if the product of their slopes is - 1.
$m_{DE}\times m_{EF}=2\times(-\frac{3}{2})=- 1$.
Since the product of the slopes of $DE$ and $EF$ is - 1, $DE$ and $EF$ are perpendicular.

Answer:

$\triangle DEF$ is a right - triangle.