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: Find the slopes of \( \overline{AB} \), \( \overline{AC} \), and \( \overline{BC} \)

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

For \( \overline{AB} \) with \( A(1, 9) \) and \( B(11, -7) \):
\( m_{AB} = \frac{-7 - 9}{11 - 1} = \frac{-16}{10} = -\frac{8}{5} \)

For \( \overline{AC} \) with \( A(1, 9) \) and \( C(-9, 3) \):
\( m_{AC} = \frac{3 - 9}{-9 - 1} = \frac{-6}{-10} = \frac{3}{5} \)

For \( \overline{BC} \) with \( B(11, -7) \) and \( C(-9, 3) \):
\( m_{BC} = \frac{3 - (-7)}{-9 - 11} = \frac{10}{-20} = -\frac{1}{2} \)

Step2: Check for perpendicular lines (product of slopes = -1)

• Check \( \overline{AB} \) and \( \overline{AC} \): \( m_{AB} \times m_{AC} = -\frac{8}{5} \times \frac{3}{5} = -\frac{24}{25}

eq -1 \)

• Check \( \overline{AB} \) and \( \overline{BC} \): \( m_{AB} \times m_{BC} = -\frac{8}{5} \times (-\frac{1}{2}) = \frac{4}{5}

eq -1 \)

• Check \( \overline{AC} \) and \( \overline{BC} \): \( m_{AC} \times m_{BC} = \frac{3}{5} \times (-\frac{1}{2}) = -\frac{3}{10}

eq -1 \)

Since no pair of sides has slopes whose product is -1, there are no perpendicular sides.

Answer:

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