Overview
What you’ll learn
- State Pythagoras’ theorem and identify the hypotenuse of a right-angled triangle.
- Calculate the hypotenuse from the two shorter sides.
- Calculate a shorter side from the hypotenuse and the other side.
- Use the converse of Pythagoras’ theorem to test whether a triangle is right-angled.
- Distinguish rational numbers from irrational numbers, and estimate surds between integers.
- Apply Pythagoras’ theorem in real-world contexts and leave exact surd answers when asked.
In 30 seconds
- 1In a right-angled triangle, the hypotenuse is the longest side, directly opposite the right angle.
- 2Pythagoras’ theorem: a² + b² = c², where c is the hypotenuse and a, b are the other two sides.
- 3Finding a shorter side rearranges the formula: a² = c² − b², then take the square root.
- 4The converse runs backwards: if a² + b² = c² holds for a triangle’s sides, the angle opposite c is a right angle.
- 5An irrational number cannot be written as a fraction of whole numbers; its decimal goes on forever without repeating.
- 6Surds like √2 and √13 are exact answers — keep them unless the question asks for a rounded decimal.
Notes · 1 of 6
The theorem
In every right-angled triangle, the squares of the short sides add up to the square of the longest.
Label a right-angled triangle so that the two sides touching the right angle are a and b, and the side opposite the right angle — the hypotenuse — is c. Pythagoras’ theorem says a² + b² = c²: the two smaller squares exactly fill the big one.
| a | b | a² + b² | c |
|---|---|---|---|
| 3 | 4 | 9 + 16 = 25 | 5 |
| 6 | 8 | 36 + 64 = 100 | 10 |
| 5 | 12 | 25 + 144 = 169 | 13 |
The theorem only applies when a right angle is present — no right angle, no deal. Whole-number triples like (3, 4, 5) and (5, 12, 13) are worth recognising on sight; exam diagrams love them.
Notes · 2 of 6
Finding the hypotenuse
Square, add, root — the standard forward trip.
- 11. Identify c
The hypotenuse is opposite the right angle; it goes alone on one side.
- 22. Square the legs
Compute a² and b² separately to keep the arithmetic visible.
- 33. Add
a² + b² = c² gives the square of the hypotenuse.
- 44. Take the root
c = √(a² + b²); leave it as a surd for an exact answer or round as instructed.
A triangle with legs 9 and 12 has c² = 81 + 144 = 225, so c = 15. With legs 5 and 7 the sum is 25 + 49 = 74, which is not a perfect square: the exact answer is √74, about 8.6. Not every triangle comes out whole — the surd is the honest form.
Notes · 3 of 6
Finding a shorter side
Subtract instead of add, then root.
When the hypotenuse is known, rearrange: from a² + b² = c², the missing leg satisfies a² = c² − b². Subtract the square of the known leg from the square of the hypotenuse — never the other way round.
| Known | Unknown | Equation | Example |
|---|---|---|---|
| both legs | hypotenuse | c² = a² + b² | 6 and 8 → c = 10 |
| hypotenuse + one leg | other leg | a² = c² − b² | 13 and 5 → a = 12 |
Notes · 4 of 6
The converse: a right-angle detector
Run the formula backwards on the three sides to test for a right angle.
The converse swaps the logic: instead of assuming a right angle, you test for one. Take the longest side as would-be c and check whether a² + b² = c². If it holds, the triangle is right-angled; if it fails, it is not — no measuring needed.
| Sides | Check | Verdict |
|---|---|---|
| 9, 12, 15 | 81 + 144 = 225 = 15² | right-angled |
| 5, 7, 9 | 25 + 49 = 74 ≠ 81 | not right-angled |
Notes · 5 of 6
Irrational numbers
Numbers that refuse to be fractions — their decimals never settle into a repeating block.
A rational number can be written as a fraction of whole numbers; its decimal either stops or repeats in a fixed cycle (√16 = 4, 0.75, 22/7 are all rational). An irrational number cannot: its decimal expansion runs forever without any repeating block. √2 is the classic — no fraction of whole numbers lands exactly on it.
| Number | Type | Why |
|---|---|---|
| 0.375 | rational | it stops: 375/1000 |
| 0.333… | rational | a fixed block (3) repeats: 1/3 |
| √9 | rational | it simplifies to exactly 3 |
| √5 | irrational | no fraction of whole numbers equals it |
Together the rationals and irrationals fill the entire number line — the real numbers. Every point is one or the other, and surds like √3 show up precisely between their rational neighbours: √3 lies between 1 and 2, since 1 = √1 < √3 < √4 = 2.
Notes · 6 of 6
Surds in the wild
Diagonals, ladders and ramps — and when to keep the root.
Real right triangles rarely come out whole. A square of side 8 has diagonal √(64 + 64) = √128 ≈ 11.3== — the surd is the exact answer and the decimal is its estimate. Keep the surd unless the question asks you to round; rounding early poisons later steps.
| Situation | Right angle hides where | Hypotenuse is |
|---|---|---|
| Ladder against a wall | wall meets ground | the ladder itself |
| TV screen diagonal | corner of the screen | the diagonal |
| Ship sailing east then north | the turning point | the straight-line distance home |
Revise
Key terms
The words examiners expect you to use precisely.
- Right-angled triangle
- A triangle containing one 90° angle.
- Hypotenuse
- The longest side of a right-angled triangle, opposite the right angle.
- Pythagoras’ theorem
- The statement a² + b² = c² linking the sides of a right-angled triangle.
- Converse
- The reversed statement: if a² + b² = c², the triangle has a right angle.
- Leg
- Either of the two shorter sides that form the right angle.
- Perfect square
- A whole number that is the square of a whole number, such as 25 or 144.
Show 6 more key termsShow fewer key terms
- Pythagorean triple
- A whole-number set of sides satisfying the theorem, such as (3, 4, 5).
- Rational number
- A number expressible as a fraction of whole numbers; its decimal stops or repeats.
- Irrational number
- A number that is not a fraction of whole numbers; its decimal never stops and never repeats.
- Surd
- An exact root left in root form, such as √2 or √13.
- Real numbers
- All rational and irrational numbers together — the whole number line.
- Exact answer
- An answer left in surd or fraction form rather than rounded.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Applying a² + b² = c² to a triangle with no right angle.
The theorem is a property of right-angled triangles only.
Instead: Check the right angle first — or use the converse when it is only claimed.
Mistake: Adding when finding a shorter side: 169 + 25 instead of 169 − 25.
The leg is missing from the sum, so its square must be subtracted from c².
Instead: a² = c² − b². A “leg” bigger than the hypotenuse is an instant red flag.
Show 8 more mistakesShow fewer mistakes
Mistake: Forgetting the final square root and answering c² as c.
The formula yields the square of the side.
Instead: After 36 + 64 = 100, one more step gives c = √100 = 10.
Mistake: Rounding √30 to 5.5 immediately and computing on with the rounded value.
Early rounding seeps into every later line and drifts from the exact answer.
Instead: Carry the surd, and round only the final answer as the question instructs.
Mistake: Calling any non-terminating decimal irrational.
Repeating decimals such as 0.333… are rational (they equal 1/3).
Instead: Irrational means no repeating block at all — like √2 or a growing pattern such as 0.101001000…
Mistake: Testing the converse with the wrong side as c.
Only the longest side can play the hypotenuse role.
Instead: Order the sides first: biggest squared should equal the sum of the other two squares.
Mistake: Refusing to leave an answer as a surd, writing √13 as 3.6 “to be safe”.
3.6 is not √13; a rounded value is an approximation, not an equality.
Instead: Exact answers stay in surd form; decimals appear only when the question says round.
Mistake: Simplifying √9 + √16 as √25.
Roots do not distribute over addition: 3 + 4 = 7, but √25 = 5.
Instead: Evaluate each root first, then add: √9 + √16 = 3 + 4 = 7.
Mistake: Assuming the product of two surds must be irrational, e.g. calling √2 × √8 irrational.
Multiply under the root: √2 × √8 = √16 = 4, a perfectly rational whole number.
Instead: Combine the roots first — the product may simplify to a rational number.
Mistake: Doubling a leg for the hypotenuse of a 45°-45°-90° triangle.
The hypotenuse satisfies h² = leg² + leg², so h = leg × √2, not 2 × leg.
Instead: Legs 5 and 5 give √50 (≈ 7.07), not 10.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Identify the hypotenuse
The hypotenuse of a right-angled triangle is
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 56 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 7 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 7.
Mark scheme
PROChapter 7 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 7 about?
Chapter 7 covers Pythagoras’ theorem and irrational numbers: the formula a² + b² = c², finding the hypotenuse or a shorter side, the converse as a right-angle test, rational versus irrational numbers, and exact surd answers.
When do I add and when do I subtract in Pythagoras problems?
Add when the hypotenuse is missing (both legs known): c² = a² + b². Subtract when a leg is missing (hypotenuse known): a² = c² − b². The hypotenuse is always the biggest number in the finished triangle.
What makes a number irrational?
It cannot be written as a fraction of whole numbers, so its decimal runs forever without a repeating block. √2, √5 and π are irrational; 0.75, 22/7 and 0.333… are rational.
Should I round √30 to a decimal?
Only if the question asks for one. Exact answers stay in surd form — √30 — and rounding early contaminates every later calculation. Bracket it between perfect squares (25 < 30 < 36) if you need a quick estimate.
Up next
Chapter 8: Trigonometric Ratios