How to Prove a Triangle is a Right Triangle Using Coordinate Geometry

To prove a triangle is a right triangle using coordinate geometry, the most direct method is to demonstrate that two of its sides are perpendicular. This is achieved by analyzing the slopes of the line segments forming the sides.

Understanding the Coordinate Geometry Approach

In the coordinate plane, the relationship between perpendicular lines is defined by their slopes. If two non-vertical lines are perpendicular, the product of their slopes is -1. This means their slopes are negative reciprocals of each other. A right triangle, by definition, contains a right angle, which is formed by two perpendicular sides.

According to geometric principles applied in a coordinate system, when proving that a triangle is a right triangle using coordinate geometry methods you must show that the slopes of two of the sides are negative reciprocals creating perpendicular lines and right angles.

Step-by-Step Method Using Slopes

Follow these steps to prove a triangle ABC is a right triangle using the slope method:

1. Identify the Vertices: Determine the coordinates of the three vertices of the triangle (A, B, and C). Let A = $(x_A, y_A)$, B = $(x_B, y_B)$, and C = $(x_C, y_C)$.
2. Calculate the Slopes: Use the slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, to find the slope of each side of the triangle.
   • Calculate the slope of side AB: $m_{AB} = \frac{y_B - y_A}{x_B - x_A}$ (if $x_B \neq x_A$).
   • Calculate the slope of side BC: $m_{BC} = \frac{y_C - y_B}{x_C - x_B}$ (if $x_C \neq x_B$).
   • Calculate the slope of side CA: $m_{CA} = \frac{y_A - y_C}{x_A - x_C}$ (if $x_A \neq x_C$).
   • Note: If any denominator is zero (e.g., $x_B - x_A = 0$), the side is a vertical line, which has an undefined slope. If one side is vertical, check if another side is horizontal (slope = 0). A vertical line and a horizontal line are perpendicular.
3. Check for Negative Reciprocal Slopes: Compare the slopes of each pair of sides (AB and BC, BC and CA, CA and AB). Multiply the slopes of each pair together.
   • If $m{AB} \times m{BC} = -1$, then side AB is perpendicular to side BC, and $\angle B$ is a right angle.
   • If $m{BC} \times m{CA} = -1$, then side BC is perpendicular to side CA, and $\angle C$ is a right angle.
   • If $m{CA} \times m{AB} = -1$, then side CA is perpendicular to side AB, and $\angle A$ is a right angle.
4. Conclusion: If you find any pair of slopes whose product is -1 (or one side is vertical and another is horizontal), you have proven that the triangle contains a right angle and is therefore a right triangle.

Example

Let's prove that the triangle with vertices A(1, 2), B(4, -2), and C(5, 5) is a right triangle.

1. Vertices: A(1, 2), B(4, -2), C(5, 5).

2. Calculate Slopes:

   • $m_{AB} = \frac{-2 - 2}{4 - 1} = \frac{-4}{3}$
   • $m_{BC} = \frac{5 - (-2)}{5 - 4} = \frac{5 + 2}{1} = \frac{7}{1} = 7$
   • $m_{CA} = \frac{2 - 5}{1 - 5} = \frac{-3}{-4} = \frac{3}{4}$

3. Check for Negative Reciprocal Slopes:

   • $m{AB} \times m{BC} = \frac{-4}{3} \times 7 = -\frac{28}{3} \neq -1$
   • $m{BC} \times m{CA} = 7 \times \frac{3}{4} = \frac{21}{4} \neq -1$
   • $m{CA} \times m{AB} = \frac{3}{4} \times \frac{-4}{3} = \frac{3 \times -4}{4 \times 3} = \frac{-12}{12} = -1$

4. Conclusion: Since $m{CA} \times m{AB} = -1$, side CA is perpendicular to side AB. This means $\angle A$ is a right angle. Therefore, triangle ABC is a right triangle.

This method effectively uses coordinate geometry principles to verify the geometric property of a right angle based on the slopes of the triangle's sides.