The Pythagorean Theorem
About 2,500 years ago, the ancient Greek mathematician Pythagoras (or his followers) proved a theorem about right triangles that still shows up in every corner of math, engineering, and science. The Pythagorean Theorem says: for any right triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides. In symbols: **a² + b² = c²**.
What is a right triangle?
A **right triangle** has one angle that measures exactly 90 degrees (a right angle — the corner of a square).\n\nThe two sides forming the right angle are called **legs** (traditionally labeled a and b).\n\nThe side OPPOSITE the right angle — always the longest side — is called the **hypotenuse** (labeled c).\n\nThe theorem only works for right triangles. Apply it to a non-right triangle, and you'll get wrong answers.
The theorem in action
Formula: **a² + b² = c²**\n\nExample: a right triangle has legs of 3 and 4. What is the hypotenuse?\n\n- 3² + 4² = c²\n- 9 + 16 = c²\n- 25 = c²\n- c = 5\n\nSo the hypotenuse is 5. This is the famous "3-4-5 right triangle" — construction workers use it to check if corners are perfectly square.
A right triangle has legs of 6 and 8. What is the length of the hypotenuse?
Finding a missing leg (CCSS 8.G.7)
The theorem works both ways. If you know the hypotenuse and one leg, you can find the other leg:\n\na² + b² = c² → a² = c² − b²\n\nExample: hypotenuse = 13, one leg = 5. Find the other leg.\n\n- a² + 5² = 13²\n- a² + 25 = 169\n- a² = 144\n- a = 12\n\nThis is another famous set — the 5-12-13 right triangle.
A ladder that is 10 feet long leans against a wall. The bottom is 6 feet from the wall. How high up the wall does the ladder reach?
Distance in the coordinate plane (8.G.8)
The Pythagorean Theorem is how we calculate distances between points on a coordinate plane. To find the distance between (x₁, y₁) and (x₂, y₂):\n\n**distance = √((x₂ − x₁)² + (y₂ − y₁)²)**\n\nThis is just the Pythagorean Theorem with the horizontal and vertical differences as the two legs, and the distance as the hypotenuse.\n\nExample: distance from (1, 2) to (4, 6)?\n\n- horizontal difference: 4 − 1 = 3\n- vertical difference: 6 − 2 = 4\n- distance = √(3² + 4²) = √25 = 5\n\nEvery GPS device on Earth uses this formula billions of times per day.
Pythagorean triples (8.G.6)
A **Pythagorean triple** is a set of three whole numbers that satisfy the theorem. The most famous:\n\n- 3, 4, 5\n- 5, 12, 13\n- 8, 15, 17\n- 7, 24, 25\n- 20, 21, 29\n\nAny multiple of a Pythagorean triple is also one: 6, 8, 10; 9, 12, 15; 30, 40, 50; etc. Ancient builders memorized these to make perfect corners without protractors. Surveyors still use them.
Measure a real right triangle
Find any right angle in the real world: a corner of a room, a book, a door frame. Measure a distance along each wall (the two "legs" of your imaginary right triangle). Then predict the diagonal using the theorem. Measure the actual diagonal to check. How close are you? Small measurement errors add up — that's engineering in practice.
Shortest path puzzle
A park is 30 m wide and 40 m long. You walk from one corner diagonally across the grass to the opposite corner. How much shorter is that than walking 30 m + 40 m = 70 m along the path? (Use the Pythagorean Theorem.) This is the real reason people cut across fields — geometry, not laziness.
Why does the Pythagorean Theorem only work for RIGHT triangles?
The Pythagorean Theorem is one of the oldest, most famous, and most useful equations in math. It connects geometry, algebra, coordinates, and distance. Every time you see a map, a building, a video game, or a GPS — it's at work. Memorize a² + b² = c². It will pay dividends for the rest of your life.
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