Question

triangle abc has the following vertices:

• a(1,9)
• b(11, - 7)
• c(-9,3)

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{ac}).
d no, because \\(\triangle abc\\) doesnt have a pair of perpendicular sides.

Explanation:

Step1: Calculate the slope of line AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A(1,9)$ and $B(11, - 7)$, $m_{AB}=\frac{-7 - 9}{11 - 1}=\frac{-16}{10}=-\frac{8}{5}$.

Step2: Calculate the slope of line BC

For points $B(11,-7)$ and $C(-9,3)$, $m_{BC}=\frac{3+7}{-9 - 11}=\frac{10}{-20}=-\frac{1}{2}$.

Step3: Calculate the slope of line AC

For points $A(1,9)$ and $C(-9,3)$, $m_{AC}=\frac{3 - 9}{-9 - 1}=\frac{-6}{-10}=\frac{3}{5}$.

Step4: Check for perpendicular - sides

Two lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1\times m_2=-1$.
$m_{AB}\times m_{AC}=-\frac{8}{5}\times\frac{3}{5}=-\frac{24}{25}
eq - 1$.
$m_{AB}\times m_{BC}=-\frac{8}{5}\times(-\frac{1}{2})=\frac{4}{5}
eq - 1$.
$m_{BC}\times m_{AC}=-\frac{1}{2}\times\frac{3}{5}=-\frac{3}{10}
eq - 1$.

Answer:

D. No, because $\triangle ABC$ doesn't have a pair of perpendicular sides.