Pythagorean Theorem: Step-by-Step Guide
In a right triangle (a triangle with a 90 degree corner), the longest side is called the hypotenuse. The Pythagorean theorem says that if you square each of the two shorter sides (the legs) and add them up, you get the square of the hypotenuse. We write this as a^2 + b^2 = c^2. If you know any two sides, you can find the third.
Math Racer skill guide · Middle School · Grades 6-8
How to solve it, step by step
- Find the legs and the hypotenuse. The hypotenuse (c) is the side across from the right angle, and it is always the longest side. The other two sides are the legs (a and b).\[a, b = \text{legs}, \quad c = \text{hypotenuse (opposite the right angle)}\]
- Write the theorem and put in the numbers. Start with a^2 + b^2 = c^2, then replace the letters you know with numbers.\[a^2 + b^2 = c^2\]
- Solve for the missing side. To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract the square of the other leg from the square of the hypotenuse and take the square root.\[c = \sqrt{a^2 + b^2},\quad a = \sqrt{c^2 - b^2}\]
- Simplify and check. Work out the square root. Check that the hypotenuse is the longest side.\[a^2 + b^2 \stackrel{?}{=} c^2\]
Worked example
A right triangle has legs of length 6 and 8. Find the length of the hypotenuse.
Step 1: The legs are 6 and 8. We need the hypotenuse, c.
Step 2: Write the theorem: a^2 + b^2 = c^2, so 6^2 + 8^2 = c^2.
Step 3: Square the legs: 36 + 64 = c^2.
Step 4: Add: c^2 = 100.
Step 5: Take the square root: c = 10.
Step 6: Check: 6^2 + 8^2 = 36 + 64 = 100, and 10^2 = 100. Correct.
The hypotenuse is 10.
Tips and common mistakes
- The theorem only works for right triangles. Make sure there is a 90 degree angle.
- The hypotenuse is always the longest side. If your answer for a leg is longer than the hypotenuse, something went wrong.
- Some sets of whole numbers work perfectly. These are called Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
- If you double or triple a triple, it still works. (6, 8, 10) is (3, 4, 5) doubled.
- When looking for a leg, you subtract: a^2 = c^2 - b^2. Do not add.
- If the square root is not a whole number, you can leave it as a square root, like the square root of 20, or use a calculator for a decimal.
- Square each side first, then add. The square root of (a^2 + b^2) is not the same as a + b.
- Draw the triangle and label the sides so you do not mix up the legs and the hypotenuse.