Geometry

To calculate the length of a side on a right-angled triangle when you know the sizes of the other two, you need to use Pythagoras' Theorem.

Pythagoras' Theorem says that, in a right angled triangle:

The square of the hypotenuse is equal to the sum of the squares on the other two sides.

Pythagoras diagram showing right angled triangle with values a, b and c and squares a≤, b≤ and c≤

We can write this more simply as :

\[{a^2} = {b^2} + {c^2}\]

Diagram of right angled triangle with a, b and c dimensions and the formula a² = b² + c²

Calculating the length of the hypotenuse

Question

Use Pythagoras' Theorem to calculate the length of the hypotenuse. Give your answer to 2 decimal places.

rectangle 4 x 7 x X

Write the equation \(x^{2} = 7^{2} + 4^{2}\)

Square the lengths you know\(x^{2} = 49 + 16\)

Add together\(x^{2} = 65\)

Find the square root\(x = \sqrt {65}\)

\[x = 8.06 (to\,2\,d.p.)\]

Question

Calculate the length of side \(x\)

(Give your answer to 2 decimal places)

Diagram of a 5 x 9 right-angled triangle with the gradient edge marked x

\[x^{2} = 5^{2} + 9^{2}\]

\[x^{2} = 25 + 81\]

\[x^{2} = 106\]

\[x = \sqrt {106}\]

\[x = 10.30 (to\, 2\,d.p.)\]

Example

Calculate the length of the side marked \(a\).

Give your answer to 2 decimal places.

rectangle 12 x 8 x a

Answer

  • Write the equation: \({12^2} = {a^2} + {8^2}\)
  • Organise the equation \({a^2} = {12^2} - {8^2}\). To find the length of a short side, we can also use the formula \({b^2} = {a^2} - {c^2}\)
  • Square the lengths you know: \({a^2} = 144 - 64\)
  • Do the subtraction: \({a^2} = 80\)
  • Find the square root: \(a = \sqrt {80}\)
  • \[a = 8.94\,(to\,2\,d.p.)\]

Question

Calculate the length of side a.

Give your answer to 2 decimal places

Right-angled triangle with values a, 9 and 13

\[a^{2} = 13^{2} - 9^{2}\]

\[a^{2} = 169 - 81\]

\[a^{2} = 88\]

\[a = \sqrt 88\]

\[a = 9.38\]