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The Pythagorean theorem
If a triangle has a right angle, its three sides are tied together by one equation. Learn it, learn four sets of numbers that fit it, and a whole family of questions becomes a few seconds of arithmetic.
In a right triangle, a² + b² = c²
The two sides that form the right angle are the legs (a and b). The longest side, across from the right angle, is the hypotenuse (c). The squares of the legs add up to the square of the hypotenuse.
Only right triangles, and c is always the longest
The theorem works only when there is a 90° angle. And c must be the side across from it: if you put a leg in the place of c, every answer comes out wrong.
How to solve them
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1
Find the hypotenuse: add the squares, take the root
Square both legs, add them, and take the square root of the total.
Next step: Find a leg- c² = 9² + 12²
- c² = 81 + 144
- c² = 225
- c = √225
- c = 15
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2
Find a leg: subtract the squares
When the hypotenuse is known, a leg is the square root of c² minus the other leg squared. Subtract, do not add.
Next step: Triples- a² = 17² − 8²
- a² = 289 − 64
- a² = 225
- a = 15
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3
Four triples worth knowing by heart
These whole-number sets fit the theorem exactly, and so does any multiple of them (6-8-10 is 3-4-5 doubled). Spot one and you can skip the arithmetic.
3-4-5and 6-8-10, 9-12-15, 30-40-50
9 + 16 = 255-12-13and 10-24-26
25 + 144 = 1698-15-17and 7-24-25 too
64 + 225 = 289Next step: Special trianglesExampleA rectangular field is 40 m by 30 m. How long is the diagonal path across it?
- The path is the hypotenuse of a triangle with legs 30 and 40.
- 30-40-50 is 3-4-5 times 10.
- The path is 50 m. Check: 900 + 1,600 = 2,500 = 50².
Answer 50 meters
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4
Two special right triangles
A 45-45-90 triangle is half a square: the legs are equal and the hypotenuse is a leg × √2. A 30-60-90 triangle is half of an equilateral triangle: the hypotenuse is twice the short leg, and the long leg is the short leg × √3.
Next step: Is it right?45-45-90A square with side 5: its diagonal
5√2 (about 7.07)
30-60-90Short leg 4
Long leg 4√3, hypotenuse 8
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5
Is it a right triangle? Test the longest side
Put the longest side as c and check whether a² + b² equals c². If it does, the triangle has a right angle.
Next step: Draw it8, 15, 1764 + 225 = 289 = 17²
Right
6, 7, 1036 + 49 = 85, but 10² = 100
Not right
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6
Word problems: draw the triangle first
Ladders against walls, wires to the ground, diagonals of rectangles and trips north then east are all right triangles. Sketch it, mark the right angle, and label which side is the hypotenuse before you calculate.
ShortcutNorth, then east
Walk 6 km north and 8 km east: the straight-line distance back to the start is the hypotenuse, and 6-8-10 says it is 10 km.
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Adding the sides
Legs 6 and 8:
Wrongc = 6 + 8 = 14
Rightc = √(36 + 64) = 10
A leg treated as the hypotenuse
Hypotenuse 13, one leg 5:
Wrong√(169 + 25)
Right√(169 − 25) = 12
Forgetting the square root
Legs 9 and 12:
Wrongc = 225
Rightc = √225 = 15
Doubling instead of × √2
A 45-45-90 triangle with legs 7:
Wronghypotenuse 14
Right7√2 ≈ 9.9
5 questions at test level
Pick an answer, then read why. No timer, no account.
A right triangle has legs of 12 and 16. How long is the hypotenuse?
Show the answer
A. 12² + 16² = 144 + 256 = 400, and √400 = 20. (It is 3-4-5 times 4.) 28 adds the legs, and 400 forgets the square root. The hypotenuse is 20.
The hypotenuse of a right triangle is 25 and one leg is 7. How long is the other leg?
Show the answer
B. 25² − 7² = 625 − 49 = 576, and √576 = 24 (the 7-24-25 triple). 18 just subtracts 25 − 7, and 26 adds the squares. The other leg is 24.
A 13-foot wire runs from the top of a pole to a stake 5 feet from the base of the pole. How tall is the pole?
Show the answer
B. The wire is the hypotenuse. 13² − 5² = 169 − 25 = 144, and √144 = 12 (the 5-12-13 triple). 13.9 adds the squares, and 8 subtracts the lengths. The pole is 12 feet tall.
Which set of side lengths forms a right triangle?
Show the answer
C. Test each with the longest side as c: 7² + 24² = 49 + 576 = 625 = 25². For 4, 5, 6: 16 + 25 = 41, not 36. For 6, 8, 11: 100, not 121. The answer is 7, 24, 25.
A square has sides of 9 inches. How long is its diagonal?
Show the answer
A. The diagonal splits the square into two 45-45-90 triangles with legs 9, so the diagonal is 9√2 (about 12.7). 18 doubles the side, and 9√3 belongs to a 30-60-90 triangle. The diagonal is 9√2.
9 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 9 cards known.
You finished the pythagorean theorem.
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