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The Pythagorean Theorem

#maths#geometry#education

What is the Pythagorean theorem? It says that in any right triangle, the square of the longest side equals the sum of the squares of the other two sides. Write the short sides as a and b and the longest side (the hypotenuse) as c, and it becomes a² + b² = c². It is the one formula that turns "I know two sides" into "I know the third", and it shows up on almost every geometry test. Here is the idea, then two examples worked all the way through.

The Pythagorean Theorema² + b² = c², alwaysa (leg)b (leg)c (hypotenuse)ACBA right triangleThe theorema² + b²=c²Square on aSquare on bSquare on c3 cm4 cmc = ?ACBThe 3-4-5 trianglePlug in the values3² + 4²=c²9 + 16=c²c²=25Solve for cc=√(25)c=5 cmc = 5 cm3 cm4 cm5 cmACBThe 3-4-5 triangle3-4-5 triangleabcACBCheck: right triangle?Reverse checkknow a, b, cif a² + b²=c²then right angle!a relationship in every right angle+ together =
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What the Pythagorean theorem says

The theorem only works for a right triangle, one with a 90° corner. The two sides that meet at that corner are the legs (a and b). The side across from the corner is the hypotenuse (c), and it is always the longest.

Why squares? Picture an actual square built on each side of the triangle. The area of the square on side a is a², on side b is b², and on side c is c². The theorem says the two smaller squares together have exactly the same area as the big one. It is a statement about area, and the formula is just that statement written down.

A worked example: the 3-4-5 triangle

Take a right triangle with legs 3 and 4. Find the hypotenuse.

  1. Write the formula: a² + b² = c²
  2. Substitute: 3² + 4² = c²
  3. Square each leg: 9 + 16 = c²
  4. Add: 25 = c²
  5. Take the square root: c = 5

The hypotenuse is 5. Check it: the squares have areas 9, 16 and 25, and 9 + 16 does equal 25. The same works for 5, 12, 13 (25 + 144 = 169) and 8, 15, 17 (64 + 225 = 289). Whole-number sets like these are called Pythagorean triples.

Finding a missing leg

Sometimes the unknown is a leg, not the hypotenuse. Then you subtract instead of add. A 10 m ladder leans against a wall with its foot 6 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse, because it is opposite the corner where wall meets ground. So c = 10 and one leg is 6.

  1. 6² + b² = 10²
  2. 36 + b² = 100
  3. b² = 100 − 36 = 64
  4. b = 8

The ladder reaches 8 m up the wall. The most common exam mistake is adding when you should subtract. Ask yourself first: is the missing side the longest one? If yes, add the squares. If no, subtract.

Using the Pythagorean theorem when the answer is not whole

Most triangles do not come out neatly. Say a rectangular screen is 16 cm wide and 9 cm tall, and you want its diagonal. The diagonal cuts the rectangle into two right triangles, so:

  • 16² + 9² = 256 + 81 = 337
  • c = √337, which is about 18.36 cm

Leave it as √337 if your teacher wants an exact answer, or round it if the question asks for decimals. A quick sanity check: the diagonal must be longer than 16 (the longest leg) but shorter than 16 + 9 = 25. Our 18.36 sits comfortably between them.

Does it work in reverse?

Yes, and this is a handy test. If three side lengths satisfy a² + b² = c², the triangle must have a right angle. Try 6, 8, 10: 36 + 64 = 100, so it is a right triangle. Try 4, 5, 7: 16 + 25 = 41, which is not 49, so it is not. Builders have long used this idea, laying out a 3-4-5 triangle with rope or a tape measure to check that a corner is square.

Quick recap

  • In a right triangle, a² + b² = c², where c is the hypotenuse, the longest side.
  • Picture it as areas: the two small squares add up to the big square.
  • Missing hypotenuse: add the squares, then take the square root.
  • Missing leg: subtract the smaller square from the hypotenuse squared, then take the square root.
  • If three sides satisfy the equation, the triangle has a right angle.

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