Year 9 · Mathematics · Measurement
Pythagoras' theorem
Use Pythagoras' theorem to find unknown side lengths in right-angled triangles and to decide whether a triangle is right-angled.
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Learning goals
Learners will
- Identify the hypotenuse of a right-angled triangle.
- Use Pythagoras' theorem to find the hypotenuse or a shorter side.
- Use the theorem to test whether a triangle contains a right angle.
Success looks like
- I can point to the hypotenuse in any right-angled triangle.
- I can find the hypotenuse given the two shorter sides.
- I can find a shorter side given the hypotenuse and the other side.
- I can solve a worded problem, such as a ladder against a wall.
The big idea
In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side. We usually call it c and call the other two sides a and b.
Pythagoras' theorem says: c^2 = a^2 + b^2. In words, if you draw a square on each side of a right-angled triangle, the area of the square on the hypotenuse equals the total area of the other two squares.
Finding the hypotenuse: square the two shorter sides, add the results, then take the square root. For sides 6 and 8: 36 + 64 = 100, and sqrt(100) = 10.
Finding a shorter side: rearrange to a^2 = c^2 − b^2. Square the hypotenuse, subtract the square of the known shorter side, then take the square root. A quick check: the answer must be shorter than the hypotenuse.
Testing for a right angle: if the square of the longest side equals the sum of the squares of the other two sides, the triangle is right-angled. If not, it is not.
Some sets of whole numbers fit the theorem exactly, such as 3, 4, 5 and 5, 12, 13. Multiples of these also work, such as 6, 8, 10. Builders have long used a 3-4-5 triangle to check that corners are square.
Worked examples
Finding the hypotenuse
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
- c^2 = 6^2 + 8^2 = 36 + 64 = 100.
- c = sqrt(100) = 10.
Answer: 10 cm
Finding a shorter side
A right-angled triangle has a hypotenuse of 13 m and one shorter side of 5 m. Find the other side.
- a^2 = c^2 − b^2 = 13^2 − 5^2 = 169 − 25 = 144.
- a = sqrt(144) = 12.
- Check: 12 is shorter than the hypotenuse of 13, which makes sense.
Answer: 12 m
Is it right-angled?
A triangle has sides 7 cm, 24 cm and 25 cm. Is it right-angled?
- The longest side is 25: 25^2 = 625.
- The other two: 7^2 + 24^2 = 49 + 576 = 625.
- The two results are equal, so the theorem holds.
Answer: Yes, it is right-angled (the right angle is opposite the 25 cm side).
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
A right-angled triangle has shorter sides of 3 cm and 4 cm. How long is the hypotenuse?
5 cm — 3^2 + 4^2 = 9 + 16 = 25, and sqrt(25) = 5 cm.
Which side of a right-angled triangle is the hypotenuse?
The side opposite the right angle — The hypotenuse is opposite the right angle and is always the longest side, whichever way the triangle is drawn.
Which set of side lengths makes a right-angled triangle?
8, 15, 17 — 8^2 + 15^2 = 64 + 225 = 289 = 17^2. The other sets fail: 36 + 49 = 85 ≠ 81, 25 + 100 = 125 ≠ 144, 16 + 25 = 41 ≠ 36.
A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm. How long is the other side?
8 cm — 10^2 − 6^2 = 100 − 36 = 64, and sqrt(64) = 8 cm.
A 5 m ladder leans against a vertical wall. Its foot is 3 m from the base of the wall on level ground. How high up the wall does it reach?
4 m — The ladder is the hypotenuse. 5^2 − 3^2 = 25 − 9 = 16, and sqrt(16) = 4 m.
A right-angled triangle has shorter sides of 9 cm and 12 cm. Find the length of the hypotenuse in cm.
15 cm — 9^2 + 12^2 = 81 + 144 = 225, and sqrt(225) = 15 cm.
A right-angled triangle has a hypotenuse of 26 m and a shorter side of 10 m. Find the other side in metres.
24 m — 26^2 − 10^2 = 676 − 100 = 576, and sqrt(576) = 24 m.
A right-angled triangle has shorter sides of 5 cm and 7 cm. Find the hypotenuse in cm, correct to one decimal place.
8.6 cm — 5^2 + 7^2 = 25 + 49 = 74, and sqrt(74) ≈ 8.602, which rounds to 8.6 cm.
Watch out for
- Adding the squares when finding a shorter side. If the hypotenuse is known, you subtract. A check: a shorter side must be less than the hypotenuse.
- Forgetting to take the square root at the end, giving c^2 instead of c.
- Thinking the hypotenuse is always the side drawn at the bottom or on the slant. Find the right angle first; the hypotenuse is directly opposite it.
- Using Pythagoras' theorem on triangles that are not right-angled. It only applies when there is a 90° angle.
Tips for parents
- Ask your learner to label the hypotenuse with a c before doing any calculation. This one habit prevents most errors.
- Encourage a quick reasonableness check: the hypotenuse must be longer than either shorter side but shorter than the two added together.
- Measure the diagonal of a rectangular table, book or TV screen together and compare it with the value from Pythagoras' theorem.
Go further
- Use string and a tape measure to mark out a 3 m, 4 m, 5 m triangle in the backyard and check that the corner is square.
- Find as many Pythagorean triads as you can with all sides less than 50. Which ones are not just multiples of another triad?
- Investigate the shortest walking route diagonally across a rectangular park compared with walking along two edges.