Pythagoras and the hypotenuse
Identify the right angle, calculate a missing side and check whether a triangle is right-angled.
Section Paper 3 · Core · Calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Square numbers and take square roots.
- Identify a right angle.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which side is the hypotenuse?
What you will learn
- Use a² + b² = c² for a right triangle.
- Subtract when finding a shorter leg and test the converse.
The hypotenuse is opposite the right angle
In a right-angled triangle, the two perpendicular legs a and b satisfy a² + b² = c², where c is the hypotenuse. It is the longest side and lies opposite the 90° angle. For legs 5 and 12 cm, c² = 25 + 144 = 169, so c = 13 cm. Take the positive square root for a physical length.
If the hypotenuse is given and a leg is missing, rearrange before substituting: a = √(c² − b²). With c = 10 and b = 6 cm, a = √64 = 8 cm. Adding squares would wrongly produce a leg longer than the hypotenuse. Keep the units consistent before squaring.
Check the condition and the result
The theorem applies only to a right triangle. To test three given side lengths, choose the longest as c and compare a² + b² with c². For 7, 24 and 25 cm, 49 + 576 = 625, so the triangle is right-angled by the converse. For 4, 5 and 6 cm, 16 + 25 ≠ 36, so it is not.
In a rectangle, a diagonal forms a right triangle with two perpendicular sides. Sketch that triangle and label the actual dimensions rather than using a sloping diagram as evidence. Leave square roots exact if requested, or round the final length to the stated accuracy, usually three significant figures for a non-exact length.
Pause and explain
Right-triangle legs are 5 and 12 cm. Find the hypotenuse.
Worked example
A right triangle has hypotenuse 10 cm and one leg 6 cm. Find the other leg.
Show the worked solution
- The missing side is a leg, so a² = 10² − 6² = 64.
- Take the positive root: a = 8 cm.
- Check 6² + 8² = 10² and that eight is shorter than ten.
Answer 8 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the hypotenuse of a right triangle with legs 8 and 15 cm.
Give me a hint
Add the squared legs before rooting.
Compare my reasoning
- c² = 64 + 225 = 289.
- c = √289 = 17 cm.
- Seventeen is longer than either leg, as a hypotenuse should be.
17 cm.
Look for these in your work
- Added squared legs.
- Took the positive square root.
A right triangle has hypotenuse 25 cm and leg 7 cm. Find the other leg.
Give me a hint
Rearrange to subtract the known leg's square.
Compare my reasoning
- a² = 625 − 49 = 576.
- a = 24 cm.
- Check 7² + 24² = 25².
24 cm.
Look for these in your work
- Subtracted for the missing leg.
- Verified the right-triangle identity.
A screen measures 9 by 12 inches. Find its diagonal and explain the angle condition.
Give me a hint
The rectangle's adjacent edges are perpendicular.
Compare my reasoning
- The diagonal forms a right triangle with legs nine and twelve.
- Its squared length is 81 + 144 = 225.
- The diagonal is 15 inches; the rectangular corner justifies Pythagoras.
15 inches.
Look for these in your work
- Identified the right triangle.
- Used the screen's perpendicular dimensions.
Common mistakes
- Using Pythagoras without a right angle.
- Adding squares for a missing leg.
- Forgetting the square root.
What can you explain now?
Do lengths 4, 5 and 6 form a right triangle?
Compare with the explanation
No.
The longest side is six, but 4² + 5² = 41 rather than 6² = 36.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, solve one hypotenuse and one leg problem, identifying c before the calculation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.