How to Use the Pythagorean Theorem to Find the Third Vertex of a Triangle

An isosceles triangle is a special kind of right triangle having vertices of 90, 45 and 45 degrees.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse -- the long side -- is equal to the sum of the squares of the other two sides. Given the length of any two sides of a right triangle, you can always use the theorem to calculate the length of the third. The angle facing the hypotenuse is by definition a 90-degree angle, and since all angles in a triangle add up to 180 degrees, the other two angles must add up to 90 degrees. To figure out the angles of those vertices requires a bit of trigonometry.

Things You'll Need

• Scientific calculator

Instructions

1. Calculate the length of the third side of the right triangle, using the Pythagorean theorem: A squared + B squared = C squared, where C is the length of the hypotenuse.

2. Assume that A and B are the lengths of the two sides that are not the hypotenuse, which is opposite the right angle. The sine of one non-right angle vertex is defined as a fraction composed of the side opposite that vertex over the hypotenuse. For the angle opposite side A, the sine is A/C.

3. Set the calculator to degrees and not radians. Punch the A/C into the calculator and calculate the answer. Tap the "Inv" key and then the "Sin" key. The result is the angle of the vertex opposite side A.

4. Subtract that angle from 90 to get the angle of the other non-right vertex.

Tips & Warnings

• If you don't have a calculator with "Inv" and "Sin" keys, you can use the online calculator at Math is Fun. Use its "asin" key.

References

• Math is Fun: Pythagoras' Theorem
• Wolfram MathWorld: Pythagorean Theorem
• Dave's Short Trig Course
• University of Guelph: Dept. of Physics: Trigonometry

Resources

• Math is Fun: Scientific Calculator