Pythagorean Theorem — Practice
In any right triangle, the two legs squared add to the hypotenuse squared. It turns a right angle into arithmetic.
What these 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.
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