Pythagoras and the hypotenuse
Identify the right angle, calculate a missing side and check whether a triangle is right-angled.
Core and Extended.
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.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources