Question

is triangle abc a right triangle? no, because none of the slopes of the line segments forming the legs are opposite reciprocals no, because none of the slopes of the line segments forming the legs are equal yes, because $overline{bc}$ is perpendicular to $overline{ac}$ yes, because $overline{ab}$ is perpendicular to $overline{ac}$ a(-5,2) b(6,5) c(4,-1)

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of $\overline{AB}$

For points $A(-5,2)$ and $B(6,5)$, $m_{AB}=\frac{5 - 2}{6-(-5)}=\frac{3}{11}$.

Step3: Calculate slope of $\overline{AC}$

For points $A(-5,2)$ and $C(4,-1)$, $m_{AC}=\frac{-1 - 2}{4-(-5)}=\frac{-3}{9}=-\frac{1}{3}$.

Step4: Calculate slope of $\overline{BC}$

For points $B(6,5)$ and $C(4,-1)$, $m_{BC}=\frac{-1 - 5}{4 - 6}=\frac{-6}{-2}=3$.

Step5: Check perpendicularity

Two lines are perpendicular if the product of their slopes is $- 1$. Since $m_{AC}\times m_{BC}=-\frac{1}{3}\times3=-1$, $\overline{AC}$ is perpendicular to $\overline{BC}$.

Answer:

yes, because $\overline{BC}$ is perpendicular to $\overline{AC}$