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 { a b } \perp \overline { a c } \\).
b yes, because \\( \overline { a b } \perp \overline { b c } \\).
c no, because \\( \overline { a b } \\) is longer than \\( \overline { a c } \\).
d no, because \\( \triangle a b c \\) doesnt have a pair of perpendicular sides.

Explanation:

Step1: Calculate the slopes of \(AB\), \(AC\), and \(BC\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \(AB\) with \(A(1,9)\) and \(B(11,-7)\):
\(m_{AB}=\frac{-7 - 9}{11 - 1}=\frac{-16}{10}=-\frac{8}{5}\)
For \(AC\) with \(A(1,9)\) and \(C(-9,3)\):
\(m_{AC}=\frac{3 - 9}{-9 - 1}=\frac{-6}{-10}=\frac{3}{5}\)
For \(BC\) with \(B(11,-7)\) and \(C(-9,3)\):
\(m_{BC}=\frac{3+7}{-9 - 11}=\frac{10}{-20}=-\frac{1}{2}\)

Step2: Check if the product of slopes is \(- 1\) (for perpendicular lines)

\(m_{AB}\times m_{AC}=(-\frac{8}{5})\times(\frac{3}{5})=-\frac{24}{25}
eq - 1\)
\(m_{AB}\times m_{BC}=(-\frac{8}{5})\times(-\frac{1}{2})=\frac{4}{5}
eq - 1\)
\(m_{AC}\times m_{BC}=(\frac{3}{5})\times(-\frac{1}{2})=-\frac{3}{10}
eq - 1\)

Answer:

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