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

To prove a triangle is a right triangle in coordinate geometry, the most common method involves examining the slopes of its sides.

You can prove a triangle is a right triangle by showing that two of its sides are perpendicular.

The Slope Method: Using Perpendicular Lines

The core principle in coordinate geometry for proving perpendicularity is related to the slopes of the lines. As the reference states, "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."

Here’s how to apply this method:

1. Find the Coordinates: Identify the coordinates of the three vertices of the triangle. Let's call them A($x_1, y_1$), B($x_2, y_2$), and C($x_3, y_3$).
2. Calculate the Slopes: Determine the slope of each side of the triangle. The slope ($m$) of a line segment connecting two points ($x_a, y_a$) and ($x_b, y_b$) is calculated using the formula:
   $m = \frac{y_b - y_a}{x_b - x_a}$
   • Calculate the slope of side AB ($m_{AB}$).
   • Calculate the slope of side BC ($m_{BC}$).
   • Calculate the slope of side AC ($m_{AC}$).
3. Check for Negative Reciprocals: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. In other words, the slope of one line is the negative reciprocal of the slope of the other line.
   • Compare $m{AB}$ and $m{BC}$. Is $m{AB} \times m{BC} = -1$?
   • Compare $m{BC}$ and $m{AC}$. Is $m{BC} \times m{AC} = -1$?
   • Compare $m{AB}$ and $m{AC}$. Is $m{AB} \times m{AC} = -1$?

If you find any pair of slopes that are negative reciprocals of each other (or whose product is -1), then the two sides forming that angle are perpendicular. This means the triangle has a right angle, proving it is a right triangle.

Handling Vertical or Horizontal Lines

Remember that the slope of a horizontal line is 0, and the slope of a vertical line is undefined. If one side is horizontal and another is vertical, they are perpendicular.

• A horizontal line has a slope of 0.
• A vertical line has an undefined slope.
• If you calculate $m{AB} = 0$ (horizontal) and $m{BC}$ is undefined (vertical), then AB is perpendicular to BC, and the triangle is a right triangle.

Example Scenario

Consider a triangle with vertices at A(1, 2), B(4, 5), and C(6, 3).

1. Slopes:
   • $m_{AB} = \frac{5 - 2}{4 - 1} = \frac{3}{3} = 1$
   • $m_{BC} = \frac{3 - 5}{6 - 4} = \frac{-2}{2} = -1$
   • $m_{AC} = \frac{3 - 2}{6 - 1} = \frac{1}{5}$
2. Check:
   • $m{AB} \times m{BC} = 1 \times (-1) = -1$.

Since the product of the slopes of AB and BC is -1, sides AB and BC are perpendicular. Therefore, the angle at vertex B is a right angle, and the triangle ABC is a right triangle.

Alternative Method: Distance Formula and Pythagorean Theorem

While the slope method is often the most direct way to show perpendicularity in coordinate geometry, you can also use the distance formula to find the length of each side and then check if the Pythagorean theorem ($a^2 + b^2 = c^2$) holds true. If the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is a right triangle.

However, as per the provided reference, the primary method to demonstrate is by showing that the slopes of two of the sides are negative reciprocals.