Pythagorean Theorem — Print
In any right triangle, the two legs squared add to the hypotenuse squared. It turns a right angle into arithmetic.
What the practice questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Base — The number being multiplied. In 53 the base is 5.
- Irrational — A real number that cannot be written as a fraction — its decimal never repeats and never ends.
- Leg — One of the two sides forming the right angle.
- Hypotenuse — The side opposite the right angle — always the longest.
- Diagonal — A segment joining two non-adjacent vertices.
- Height — The perpendicular distance from the base to the opposite corner.
Draw a right triangle with legs 3 and 4. Build a square on each of the three sides.
Count the little tiles in each square: 9 on one leg, 16 on the other.
Now count the square on the long slanted side. It holds exactly 25.
And 9 + 16 = 25. Not approximately — exactly, every time, for every right triangle.
Try it with legs 5 and 12: the squares hold 25 and 144, and the big one holds 169.
That is what a2 + b2 = c2 is saying. The letters look like algebra, but the statement is about areas of real squares — which is why the theorem is true rather than merely useful.
The statement, and what the letters mean
a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.
c is always the side opposite the right angle, and always the longest. Getting this wrong is the main source of errors.
The theorem only applies to right triangles. No right angle, no formula.
Finding the hypotenuse
Legs 6 and 8: 62 + 82 = 36 + 64 = 100, so c = √100 = 10.
Square both legs, add, then take the square root. Three steps, in that order.
Sanity check
The hypotenuse must be longer than either leg, but shorter than their sum. For legs 6 and 8, expect between 8 and 14. A "hypotenuse" of 5 is impossible.
Finding a leg — subtract, do not add
If you know the hypotenuse, rearrange: a2 = c2 - b2.
Hypotenuse 13, one leg 5: a2 = 169 - 25 = 144, so a = 12.
Adding here instead of subtracting is the most common slip in this unit. Identify the hypotenuse first, and the direction follows.
Triples worth memorising
3-4-5, 5-12-13, 8-15-17, 7-24-25.
Multiples count too: 6-8-10 and 9-12-15 are both just 3-4-5 scaled.
Spotting one saves all the arithmetic — and catching that 10-24-26 is 2 × (5-12-13) is faster than squaring anything.
The converse: is it a right triangle?
Test the three sides. If a2 + b2 = c2 with c the longest, the triangle is right-angled.
9, 12, 15: 81 + 144 = 225 = 152 ✓ — right-angled.
4, 5, 8: 16 + 25 = 41 but 64 — not right-angled. Since 41 < 64, the angle opposite 8 is obtuse.
Where it actually gets used
The diagonal of a rectangle, the height of a ladder against a wall, the straight-line distance between two points on a grid.
On coordinates, the horizontal and vertical gaps are the legs, and the distance between the points is the hypotenuse.
Worked examples
Find the hypotenuse
A right triangle has legs 9 and 12. Find the hypotenuse.
- 92 + 122 = 81 + 144 = 225 Square each leg and add.
- c = √225 = 15 Take the root of the total.
- Check: 15 > 12 and 15 < 21 ✓ Between the longest leg and the sum — plausible. Also 3 × (3-4-5).
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Find a leg
A ladder 17 ft long leans against a wall, its foot 8 ft out. How high up the wall does it reach?
- The ladder is the hypotenuse: c = 17. It is opposite the right angle between wall and ground.
- a2 = 172 - 82 = 289 - 64 = 225 Subtract, because you are finding a leg.
- a = √225 = 15 An 8-15-17 triple.
15 feet up the wall.
Distance on a grid
Find the distance between (2, 3) and (7, 15).
- Horizontal gap: 7 - 2 = 5 One leg of a right triangle.
- Vertical gap: 15 - 3 = 12 The other leg.
- 52 + 122 = 25 + 144 = 169 Apply the theorem to the two gaps.
- √169 = 13 A 5-12-13 triple.
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Words to know
- Leg
- One of the two sides forming the right angle.
- Hypotenuse
- The side opposite the right angle — always the longest.
- Pythagorean triple
- Three whole numbers that fit: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
- Converse
- If a2 + b2 = c2, then the triangle must be right-angled.
Mistakes to watch for
Treating a leg as the hypotenuse. c is always opposite the right angle and always the longest side. Put the longest length in the wrong place and the arithmetic gives a leg that is longer than the hypotenuse — impossible.
Adding when you should subtract. If you know the hypotenuse and one leg, the missing leg comes from c2 - a2. Adding gives an answer bigger than the hypotenuse, which the picture immediately disproves.
Forgetting the final square root. 9 + 16 = 25 is c2, not c. Stopping there gives an answer five times too large and still looks plausible.
Using it on a triangle with no right angle. The whole proof rests on that corner. Without a right angle the relationship simply does not hold.
The theorem has more known proofs than any other in mathematics — one 1927 collection gathered 370 of them.
One was published in 1876 by James A. Garfield, five years before he became President of the United States. His proof uses a trapezium and is genuinely elegant.
Babylonian tablets list Pythagorean triples from around 1800 BC, a thousand years before Pythagoras. What the Greeks added was the proof that it always holds.
Problem 1
A right triangle has legs 3 and 4. Find the hypotenuse.
Worked answer
5
9 + 16 = 25
c = √25 = 5
Problem 2
A right triangle has legs 6 and 8. Find the hypotenuse.
Worked answer
10
36 + 64 = 100
c = 10
Problem 3
A right triangle has legs 5 and 12. Find the hypotenuse.
Worked answer
13
25 + 144 = 169
c = 13
Problem 4
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
Worked answer
12
169 - 25 = 144
√144 = 12
Problem 5
Which side of a right triangle is the hypotenuse?
- The shortest side
- The side opposite the right angle
- Either of the two legs
Worked answer
The side opposite the right angle
The hypotenuse is opposite the right angle, and always the longest side.
Problem 6
A right triangle has legs 9 and 12. Find the hypotenuse.
Worked answer
15
81 + 144 = 225
c = 15
Problem 7
A 17 ft ladder leans against a wall with its foot 8 ft from the base. How high up the wall does it reach?
Worked answer
15
289 - 64 = 225
√225 = 15 ft
Problem 8
A right triangle has legs 8 and 15. Find the hypotenuse.
Worked answer
17
64 + 225 = 289
c = 17
Problem 9
A right triangle has hypotenuse 25 and one leg 7. Find the other leg.
Worked answer
24
625 - 49 = 576
√576 = 24
Problem 10
Is a triangle with sides 9, 12 and 15 a right triangle?
- Yes
- No
Worked answer
Yes
81 + 144 = 225
152 = 225 ✓
Equal, so it is right-angled.
Problem 11
Is a triangle with sides 4, 5 and 8 a right triangle?
- Yes
- No
Worked answer
No
16 + 25 = 41
82 = 64
41 ≠ 64, so it is not right-angled.
Problem 12
A rectangle measures 6 by 8. Find the length of its diagonal.
Worked answer
10
Legs 6 and 8
36 + 64 = 100, so the diagonal is 10.
Problem 13
Find the distance between the points (2, 3) and (7, 15).
Worked answer
13
Horizontal: 7 - 2 = 5
Vertical: 15 - 3 = 12
25 + 144 = 169, so the distance is 13.
Problem 14
A right triangle has both legs equal to 1. What is its hypotenuse?
- 1
- 2
- √2
- √3
Worked answer
√2
12 + 12 = 2
c = √2 ≈ 1.414 — an irrational number.
Problem 15
A square has side 10. Find its diagonal to the nearest tenth.
Worked answer
14.1
100 + 100 = 200
√200 ≈ 14.14
≈ 14.1
Problem 16
A 20 ft ladder reaches 16 ft up a wall. How far is its foot from the wall?
Worked answer
12
400 - 256 = 144
√144 = 12 ft