Watch out
Using Pythagoras on a triangle that is not right-angled.
Scottish National 5 Mathematics
National 5 Maths Pythagoras
Apply Pythagoras in complex, converse and three-dimensional situations.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Identify the right angle.
- Label the hypotenuse as the longest side, opposite the right angle.
- Decide whether you are finding the hypotenuse or a shorter side.
Explanation
Pythagoras' theorem works only in right-angled triangles. It connects the two shorter sides and the hypotenuse.
If you are finding the hypotenuse, add the squares of the two shorter sides. If you are finding a shorter side, subtract the known shorter-side square from the hypotenuse square.
Always square the lengths first, then take the square root at the end.
Visual support
Right-angled triangle
Key formulae and rules
- a² + b² = c²
- c is the hypotenuse
- shorter side² = c² − other shorter side²
Check
The hypotenuse is opposite the right angle and must be the longest side of the right-angled triangle.
Exam tip
State which side is the hypotenuse before substituting. This prevents adding when you should subtract.
Worked examples
Worked example 1
Find the hypotenuse
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
- c² = 6² + 8²
- c² = 36 + 64 = 100
- c = √100 = 10
Answer: The hypotenuse is 10 cm.
Worked example 2
Find a shorter side
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other shorter side.
- x² + 5² = 13²
- x² + 25 = 169
- x² = 144
- x = 12
So: The missing side is 12 cm.
Worked example 3
Use Pythagoras in context
A ladder reaches 4.8 m up a wall and its foot is 1.4 m from the wall. Find the ladder length.
- The ladder is the hypotenuse.
- l² = 4.8² + 1.4²
- l² = 23.04 + 1.96 = 25
- l = 5
Answer: The ladder is 5 m long.
Watch out
- Using Pythagoras on a triangle that is not right-angled.
- Adding when the missing side is not the hypotenuse.
- Forgetting to take the square root at the end.