Geometry & Measures
Grade 5-7Pythagoras' Theorem
Pythagoras theorem connects the three sides of any right-angled triangle, and GCSE questions test whether you can find a missing side in either direction, check whether a triangle is right-angled from its three side lengths, and apply the same idea to points on a coordinate grid. This lesson covers finding the hypotenuse, finding a shorter side, testing for a right angle, and using the theorem with coordinates and surds.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the hypotenuse
In a right-angled triangle, the hypotenuse (the longest side, opposite the right angle) squared equals the sum of the squares of the other two sides: a squared plus b squared equals c squared, where c is the hypotenuse. To find the hypotenuse, square both shorter sides, add them together, then take the square root.
A right-angled triangle has shorter sides of 6 cm and 8 cm. Since 6 squared plus 8 squared equals 36 plus 64, which is 100, the hypotenuse is the square root of 100, which is 10 cm.
Finding a shorter side
If the hypotenuse and one shorter side are known, rearrange the theorem: square both known sides, subtract the smaller square from the larger, then take the square root of the result.
A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Since 13 squared minus 5 squared equals 169 minus 25, which is 144, the missing side is the square root of 144, which is 12 cm.
Testing whether a triangle is right-angled
Given three side lengths, a triangle is right-angled only if the square of the longest side exactly equals the sum of the squares of the other two. If the two sides of the equation do not match, the triangle is not right-angled.
A triangle has sides 7 cm, 8 cm and 12 cm. Since 7 squared plus 8 squared equals 49 plus 64, which is 113, but 12 squared is 144, and 113 does not equal 144, the triangle is not right-angled.
Coordinates and surds
The distance between two points on a coordinate grid can be found by treating the horizontal and vertical differences as the two shorter sides of a right-angled triangle, then applying Pythagoras theorem. Answers are sometimes left as a surd rather than a decimal, particularly when an exact value is required.
The distance between the points (1, 2) and (7, 10) uses a horizontal difference of 6 and a vertical difference of 8, giving a distance of the square root of 6 squared plus 8 squared, which is the square root of 100, or 10 units. A square with sides of 6 cm has a diagonal of the square root of 6 squared plus 6 squared, which is the square root of 72, simplifying to 6 root 2 cm.
Worked Examples
Three exam-style questions, fully solved.
Find the length of the hypotenuse of a right-angled triangle with shorter sides 6 cm and 8 cm.
Easy- 1.Square both shorter sides and add: 6² + 8² = 100
- 2.Take the square root to find the hypotenuse
Answer: x = 10 cm
A triangle has sides of length 9 cm, 12 cm and 15 cm. Determine whether the triangle is right-angled.
Medium- 1.Square the two shorter sides and add them: 9² + 12² = 81 + 144 = 225
- 2.Square the longest side: 15² = 225
- 3.Compare the two results
Answer: Since 9² + 12² = 15², the triangle is right-angled
A right-angled triangle has legs of exact length root 8 cm and root 28 cm. Find the exact length of the hypotenuse, simplified as far as possible.
Hard- 1.Square each leg: (root 8)² = 8 and (root 28)² = 28
- 2.Add the squared values: 8 + 28 = 36
- 3.Take the square root of the total
Answer: h = 6 cm
Avoid These
The most common mistakes students make.
Adding the squares of two sides when one of them is actually the hypotenuse, instead of subtracting to find a shorter side.
Forgetting to take the square root at the end, leaving the answer as the squared value instead of the actual length.
When testing whether a triangle is right-angled, comparing the wrong pair of values, such as squaring the two longer sides instead of the two shorter sides.
Mixing up horizontal and vertical differences when finding a distance between two coordinates, or forgetting to subtract the coordinates in the correct order before squaring.
Not simplifying a surd answer fully, such as leaving root 72 instead of simplifying it to 6 root 2.
FAQ
Questions parents and students ask.
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