How to Prove a Triangle Is a Right Triangle With Vertices

Only right triangle's possess a hypotenuse.

A triangle's type depends on its interior angles. A right triangle is named for its right angle, which measures 90 degrees. The vertices of a triangle are the points that make up its corners. How those vertices are connected determines the length of the triangle's sides and the measurements of its interior angles. With the three vertices, which also are Cartesian points, you can determine whether a triangle is a right triangle. You find the squares of the distances between the vertices and substitute them into the Pythagorean theorem.

Instructions

1. Label the three vertices as P1, P2 and P3 to help with identification. For this example, the three vertices are (2,3), (5,5), and (4,0). Let P1 be (2, 3), P2 be (5, 5) and P3 be (4, 0).

2. Subtract P1's x-coordinate from P2's. Square the difference. Subtract P1's y-coordinate from P2's. Square the difference. Add the squares together. For this example, 5 - 2 equals 3, which squared is 9; 5 - 3 is 2, which squared is 4; and 9 added to 4 is 13.

3. Subtract P2's x-coordinate from P3's. Square the difference. Subtract P2's y-coordinate from P3's. Square the difference. Add the squares together. For this example, 4 - 5 equals -1, which squared is 1; 0 - 5 is -5, which squared is 25; and 25 added to 1 is 26.

4. Subtract P3's x-coordinate from P1's. Square the difference. Subtract P3's y-coordinate from P1's. Square the difference. Add the squares together. For this example, 2 - 4 equals -2, which squared is 4; 3 - 0 is 3, which squared is 9; and 9 added to 4 is 13.

5. Add the two smaller sums from the previous three steps to see if they add up to the largest sum. If so, then the three sides satisfy the Pythagorean theorem and the three vertices construct a right triangle. The Pythagorean theorem declares that c^2 = a^2 + b^2, where c is the hypotenuse --- the side opposite the right angle -- and a and b are the other two sides. Concluding this example, the sums 13 and 13 are smaller than 26, and adding 13 to 13 equals 26; the three vertices are those of a right triangle.

Tips & Warnings

• If you have a protractor, connect the three vertices together and measure each of the three angles created. If one of the angles measures 90 degrees, then the triangle is a right triangle.

References

• Wolfram MathWorld: Pythagorean Theorem
• Purplemath: The Distance Formula