Pythagoras' Theorem
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is Pythagoras' Theorem?
For any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².
a² + b² = c²: the areas of the two smaller squares add up to the area of the largest square
- The hypotenuse is always the longest side, and it is always opposite the right angle - not necessarily the "bottom" side in a diagram.
- Pythagoras' theorem only works for right-angled triangles.
Finding the Hypotenuse
To find the hypotenuse, add the squares of the two shorter sides, then take the square root.
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Question: A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.
- Step 1: c² = 6² + 8² = 36 + 64 = 100
- Step 2: c = √100
- Answer: c = 10 cm
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Question: A right-angled triangle has legs 5 cm and 12 cm. Find the hypotenuse.
- Step 1: c² = 5² + 12² = 25 + 144 = 169
- Step 2: c = √169
- Answer: c = 13 cm
Finding a Shorter Side
To find a shorter side when the hypotenuse is known, subtract the known leg's square from the hypotenuse's square, then take the square root.
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Question: A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
- Step 1: b² = 13² − 5² = 169 − 25 = 144
- Step 2: b = √144
- Answer: b = 12 cm
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Question: A right-angled triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.
- Step 1: b² = 17² − 8² = 289 − 64 = 225
- Step 2: b = √225
- Answer: b = 15 cm
Checking Whether a Triangle is Right-Angled
The converse of Pythagoras' theorem: if a² + b² = c² is true for a triangle's three sides (with c the longest), the triangle must be right-angled.
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Question: A triangle has sides 7 cm, 24 cm, and 25 cm. Is it right-angled?
- Step 1: 7² + 24² = 49 + 576 = 625
- Step 2: 25² = 625
- Answer: Yes - 625 = 625, so it is right-angled
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Question: A triangle has sides 5 cm, 6 cm, and 8 cm. Is it right-angled?
- Step 1: 5² + 6² = 25 + 36 = 61
- Step 2: 8² = 64
- Answer: No - 61 ≠ 64, so it is not right-angled
Pythagoras in Real-World Problems
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Question: A ladder 10 m long leans against a vertical wall, with its foot 6 m from the base of the wall. How high up the wall does the ladder reach?
- Step 1: The ladder is the hypotenuse: height² = 10² − 6² = 100 − 36 = 64
- Step 2: height = √64
- Answer: 8 m
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Question: A rectangular field is 40 m long and 30 m wide. Find the length of the diagonal path across the field.
- Step 1: diagonal² = 40² + 30² = 1600 + 900 = 2500
- Step 2: diagonal = √2500
- Answer: 50 m
Pythagoras as Part of a Larger Problem
Pythagoras' theorem is often just ONE step inside a bigger question, such as finding a compound shape's perimeter or area - spot where a right-angled triangle is hiding, use Pythagoras to find its missing side, then continue with the rest of the question.
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Question: A field's boundary has three straight sides of 15 m, 22 m, and 18 m, plus one diagonal side connecting the last two corners. The diagonal has a horizontal run of 20 m and a vertical drop of 21 m. Find the total perimeter of the field.
- Step 1 (find the diagonal using Pythagoras): diagonal² = 20² + 21² = 400 + 441 = 841
- Step 2: diagonal = √841 = 29 m
- Step 3 (now finish the original question): Total perimeter = 15 + 22 + 18 + 29
- Answer: 84 m
Real-World Applications
Pythagoras' theorem is used in many practical situations:
- Construction: checking corners are exact right angles using the 3-4-5 rule.
- Ladder safety: calculating safe height and distance from a wall.
- Navigation: finding the shortest (diagonal) distance between two points.
- Sports field marking: checking diagonal measurements are correct.
- Screen sizes: diagonal measurement of TVs and monitors from width and height.
Exam Tips
Common Mistakes
MistakeAdding when finding a shorter side, e.g. b² = c² + a² instead of b² = c² − a²
Fixto find a shorter side, SUBTRACT the known leg's square from the hypotenuse's square
MistakeForgetting to take the square root at the end, leaving the answer squared
Fixafter finding c² (or a² or b²), always take the square root to get the actual side length
MistakeMisidentifying which side is the hypotenuse in a rotated or unusually drawn triangle
Fixthe hypotenuse is always the longest side, opposite the right angle
MistakeApplying Pythagoras' theorem to a triangle that isn't right-angled
FixPythagoras' theorem only applies to right-angled triangles
MistakeUsing the wrong pair of sides when checking if a triangle is right-angled
Fixsquare the two SHORTER sides and add them, then compare to the LONGEST side squared
MistakeStopping after finding a missing side with Pythagoras, without finishing the rest of the question
FixPythagoras is often just one step - check whether the question also wants a perimeter, area, or other total
For Exams
- Learn the formula: a² + b² = c², where c is always the hypotenuse.
- Finding the hypotenuse: ADD the squares of the shorter sides, then square root.
- Finding a shorter side: SUBTRACT the known square from the hypotenuse's square, then square root.
- Identify the hypotenuse first, before starting any calculation.
- Watch for Pythagoras hiding inside a bigger problem, like a compound shape's perimeter or area.
- Learn common triples: 3-4-5, 5-12-13, 8-15-17 - they appear often in exams.
Interactive revision notes, videos and practice questions load below.