Question

triangle abc has the following vertices:

• a(-8,7)
• b(0,-5)
• c(9,1)

is triangle abc a right triangle, and why?
choose 1 answer:
a yes, because \\(\overline{ab}\perp\overline{ac}).
b yes, because \\(\overline{ab}\perp\overline{bc}).
c no, because \\(\overline{ab}\\) is longer than \\(\overline{bc}).
d no, because \\(\triangle abc\\) doesnt have a pair of perpendicular sides.

Explanation:

Step1: Calculate slope of line AB

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-8,7)$ and $B(0, - 5)$, $m_{AB}=\frac{-5 - 7}{0-(-8)}=\frac{-12}{8}=-\frac{3}{2}$.

Step2: Calculate slope of line BC

For points $B(0,-5)$ and $C(9,1)$, $m_{BC}=\frac{1-(-5)}{9 - 0}=\frac{6}{9}=\frac{2}{3}$.

Step3: Calculate slope of line AC

For points $A(-8,7)$ and $C(9,1)$, $m_{AC}=\frac{1 - 7}{9-(-8)}=\frac{-6}{17}$.

Step4: Check for perpendicular - lines

Two lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1\times m_2=-1$. Since $m_{AB}\times m_{BC}=-\frac{3}{2}\times\frac{2}{3}=-1$, $\overline{AB}\perp\overline{BC}$.

Answer:

B. Yes, because $\overline{AB}\perp\overline{BC}$.