To show that triangle ABC is a right triangle, we need to demonstrate that one of its angles is a right angle (90°). A triangle is a right triangle if the square of the length of one side equals the sum of the squares of the other two sides (Pythagoras' theorem), or if two sides are perpendicular, which can be determined by checking if the slopes of two sides are negative reciprocals.

Steps:

1. Calculate the slopes of the sides AB, BC, and AC:

   The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

   $m = \frac{y_2 - y_1}{x_2 - x_1}$

   • Slope of AB: $m_{AB} = \frac{7 - 5}{-5 - 2} = \frac{2}{-7} = -\frac{2}{7}$

   • Slope of BC: $m_{BC} = \frac{-9 - 7}{-2 - (-5)} = \frac{-16}{3}$

   • Slope of AC: $m_{AC} = \frac{-9 - 5}{-2 - 2} = \frac{-14}{-4} = \frac{7}{2}$

2. Check if any two sides are perpendicular:

   Two lines are perpendicular if their slopes are negative reciprocals, meaning $m_1 \cdot m_2 = -1$.

   • Check slopes of AB and AC: $m_{AB} \cdot m_{AC} = -\frac{2}{7} \cdot \frac{7}{2} = -1$ Since the product of the slopes is $-1$, sides AB and AC are perpendicular, meaning $\angle A$ is a right angle.

Conclusion:

Since AB is perpendicular to AC, $\triangle ABC$ is a right triangle, with a right angle at point A.

Would you like further details or have any questions about the process?

Related Questions:

1. How do you find the equation of a line given two points?
2. Can you prove a triangle is isosceles using slopes?
3. How do you find the length of a line segment between two points?
4. How can you find the area of a triangle using coordinates?
5. How do you determine if three points are collinear?

Tip:

Remember, two lines are perpendicular when their slopes multiply to $-1$, which is a quick way to check for right angles in geometric problems.