Question

triangle abc has vertices at a(-2, 3), b(-3, -6), and c(2, -1). is triangle abc a right triangle? if so, which angle is the right angle?
no, the triangle has no right angles.
yes, the right angle is angle a.
yes, the right angle is angle b.
yes, the right angle is angle c.

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For \(AB\): \(m_{AB}=\frac{-6 - 3}{-3+2}=\frac{-9}{-1} = 9\).
For \(BC\): \(m_{BC}=\frac{-1+6}{2 + 3}=\frac{5}{5}=1\).
For \(AC\): \(m_{AC}=\frac{-1 - 3}{2+2}=\frac{-4}{4}=-1\).

Step2: Check for perpendicular slopes

Two lines are perpendicular if \(m_1\times m_2=-1\).
Since \(m_{BC}\times m_{AC}=1\times(-1)=-1\), \(BC\perp AC\).

Answer:

Yes, the right angle is angle \(C\).