Geometry & measures
Pythagoras' Theorem
Year 8 · Year 9
- ✓By the end of this lesson students will be able to identify a right-angled triangle and its hypotenuse.
- ✓By the end of this lesson students will be able to state Pythagoras' theorem.
- ✓By the end of this lesson students will be able to calculate the length of the hypotenuse in a right-angled triangle.
- ✓By the end of this lesson students will be able to calculate the length of a shorter side in a right-angled triangle.
- ✓By the end of this lesson students will be able to solve problems involving Pythagoras' theorem in two dimensions.
Key concepts
A right-angled triangle is a triangle that contains one angle exactly equal to 90 degrees. This angle is often marked with a square symbol.
The hypotenuse is the longest side of a right-angled triangle. It is always the side directly opposite the right angle.
Pythagoras' theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. If 'a' and 'b' are the lengths of the two shorter sides, and 'c' is the length of the hypotenuse, the theorem can be written as:
The two sides of a right-angled triangle that form the right angle are called the shorter sides or legs. In the formula a² + b² = c², these are represented by 'a' and 'b'.
Key facts to remember
- 1Pythagoras' theorem only applies to right-angled triangles.
- 2The hypotenuse is always the longest side of a right-angled triangle and is opposite the right angle.
- 3The formula is a² + b² = c², where 'c' is the hypotenuse and 'a' and 'b' are the other two shorter sides.
- 4To find the hypotenuse, you add the squares of the shorter sides: c = √(a² + b²).
- 5To find a shorter side, you subtract the square of the known shorter side from the square of the hypotenuse: a = √(c² - b²) or b = √(c² - a²).
- 6Always remember to take the square root as the final step.
- 7Units must be consistent and included in your final answer.
Worked examples
Example 1
A right-angled triangle has two shorter sides of lengths 5 cm and 12 cm. Calculate the length of the hypotenuse.
Answer
c = 13 cm
The hypotenuse is the longest side, and 13 cm is indeed longer than 5 cm and 12 cm, so the answer makes sense.
Example 2
A right-angled triangle has a hypotenuse of 10 m and one shorter side of 6 m. Calculate the length of the other shorter side.
Answer
b = 8 m
The shorter side must be less than the hypotenuse (10 m), and 8 m is indeed less than 10 m.
Example 3
A ladder of length 7.5 m leans against a vertical wall. The base of the ladder is 2.1 m away from the wall. How high up the wall does the ladder reach? Give your answer to 3 significant figures.
Answer
The ladder reaches 7.20 m up the wall (to 3 significant figures).
Ensure you round to the specified number of significant figures or decimal places. In this case, 7.2 is 7.20 to 3 s.f.
Common mistakes
- ✗Applying Pythagoras' theorem to triangles that are not right-angled.
- ✗Incorrectly identifying the hypotenuse (it's always opposite the right angle and the longest side).
- ✗Adding the squares of the sides when trying to find a shorter side (it should be subtraction).
- ✗Forgetting to take the square root at the end of the calculation, leaving the answer as c² or a².
- ✗Making calculation errors with squares or square roots, especially with non-integer values.
Exam tips
- ★Always draw a clear diagram and label the sides, especially 'c' for the hypotenuse.
- ★Write down the formula a² + b² = c² at the start of your working.
- ★Show all your steps clearly, including the substitution of values and the square root operation.
- ★Check if your answer makes sense: the hypotenuse must be the longest side, and a shorter side must be shorter than the hypotenuse.
- ★Pay close attention to rounding instructions in the question (e.g., to 1 decimal place, 3 significant figures).
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