Pythagoras' theorem · Year 9
In Pythagoras' theorem (Year 9) you practise 3 skills: Finding the hypotenuse and a shorter side · Pythagoras' theorem in shapes · Is it a right triangle?. In a right triangle, the side opposite the right angle is called the hypotenuse. Pythagoras' theorem says: the squares of the two shorter sides add up to the square of the hypotenuse.
The theorem helps you find lengths in shapes: the diagonal of a rectangle is the hypotenuse of a triangle whose shorter sides are the sides of the rectangle. The height of an isosceles triangle makes a right triangle too. Sometimes the hypotenuse is not a whole number. For example, if the shorter sides are 5 and 6, the hypotenuse squared is 61. The square root of 61 is between 7 and 8, and it is about 7.8. The theorem also works the other way round: if the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. That is how you can make a right angle with a rope: 3 metres, 4 metres and 5 metres.
Tip: The hypotenuse is always the longest side. If your hypotenuse comes out shorter than one of the other sides, check your working.
What you learn
Finding the hypotenuse and a shorter side
For example: In a right triangle, the two shorter sides are 5 and 12. How long is the hypotenuse?
Answer and solution
The answer: 13
5² + 12² = 25 + 144 = 169 The hypotenuse squared is 169, so the hypotenuse is the square root of 169: √169 = 13
Pythagoras' theorem in shapes
For example: The sides of a rectangle are 12 cm and 9 cm. How long is its diagonal?
Answer and solution
The answer: 15 cm
By Pythagoras' theorem: 12² + 9² = 144 + 81 = 225 The diagonal squared is 225, so: √225 = 15 The diagonal is 15 cm long.
Is it a right triangle?
For example: The sides of a triangle are 6, 12 and 8. Does the triangle have a right angle?
- Yes
- No
Answer and solution
The answer: No
Check: the sum of the squares of the two shorter sides, against the square of the longest side: 6² + 8² = 36 + 64 = 100 12² = 144 100 is not equal to 144, so the triangle has no right angle.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).