Pythagorean Theorem — Challenge
In any right triangle, the two legs squared add to the hypotenuse squared. It turns a right angle into arithmetic.
What these problems 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.
Problem 1
A right triangle has legs of 3 and 4. What is the length of its hypotenuse?
- 4
- 5
- 6
- 7
- 25
Worked answer
5
Pythagoras' theorem says the square of the hypotenuse equals the sum of the squares of the legs.
32+42 = 9+16 = 25, so the hypotenuse is √25 = 5.
Forgetting the square root leaves 25, which is the squared length.
Problem 2
A right triangle has legs of 6 and 8. What is the length of its hypotenuse?
- 7
- 9
- 10
- 14
- 48
Worked answer
10
62+82 = 36+64 = 100.
The hypotenuse is √100 = 10.
Adding the legs gives 14, which is always too large for a hypotenuse.
Problem 3
A right triangle has a hypotenuse of 13 and one leg of 5. How long is the other leg?
- 8
- 10
- 12
- 14
- 18
Worked answer
12
Here the hypotenuse is known, so subtract rather than add: 132-52.
169-25 = 144, so the other leg is √144 = 12.
Adding the squares would give a length longer than the hypotenuse, which is impossible.
Problem 4
A right triangle has legs of 5 and 12. What is the length of its hypotenuse?
- 7
- 10
- 12
- 13
- 17
Worked answer
13
52+122 = 25+144 = 169.
The hypotenuse is √169 = 13.
Problem 5
A right triangle has both legs equal to 1. What is the length of its hypotenuse?
- √2
- 1
- 2
- √3
- 4
Worked answer
√2
12+12 = 2, so the hypotenuse is √2.
This length is irrational, about 1.414, and cannot be written as a fraction.
Problem 6
Which of these sets of three lengths is not the side lengths of a right triangle?
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- 9, 12, 15
- 6, 8, 11
Worked answer
6, 8, 11
A triangle is right-angled exactly when the squares of the two shorter sides add to the square of the longest.
Checking each: 25+144=169, 64+225=289, 49+576=625 and 81+144=225 all work.
For 6, 8, 11: 36+64 = 100 but 112 = 121, so this set is not right-angled.
Problem 7
A right triangle has legs of 8 and 15. What is the length of its hypotenuse?
- 12
- 15
- 17
- 23
- 120
Worked answer
17
82+152 = 64+225 = 289.
The hypotenuse is √289 = 17.
Problem 8
A ladder 13 metres long leans against a wall with its foot 5 metres from the wall. How high up the wall does it reach, in metres?
- 8
- 10
- 12
- 14
- 18
Worked answer
12
The ladder is the hypotenuse and the distance from the wall is one leg.
The height is √132-52 = √169-25 = √144 = 12 metres.
Problem 9
What is the length of the diagonal of a rectangle measuring 9 by 12?
- 12
- 15
- 18
- 21
- 225
Worked answer
15
The diagonal cuts the rectangle into two right triangles with legs 9 and 12.
92+122 = 81+144 = 225, so the diagonal is √225 = 15.
Problem 10
What is the length of the diagonal of a square with side 6?
- 6
- 12
- 6√2
- 3√2
- 36
Worked answer
6√2
The diagonal is the hypotenuse of a right triangle with both legs equal to 6.
62+62 = 72, so the diagonal is √72 = 6√2.
In general a square of side s has diagonal s√2, about 1.414 times the side.