Year 10: Pythagoras' theorem

Pythagoras' theorem is one of the most important geometry skills in Australian Year 10. It works in right-angled triangles and lets you connect the two shorter sides with the longest side, called the hypotenuse.

In a right-angled triangle,

a^{2} + b^{2} = c^{2}

, where

c

is the hypotenuse.

Recognising when to use it

Use Pythagoras only when the triangle has a right angle. The hypotenuse must be opposite the right angle and is always the longest side.

Example 1: A right-angled triangle has shorter sides of

6 cm

and

8 cm

. Find the hypotenuse.

Write the formula: $a^{2} + b^{2} = c^{2}$ .
Substitute: $6^{2} + 8^{2} = c^{2}$ .
Calculate: $36 + 64 = 100$ .
So $c^{2} = 100$ , giving $c = 10$ .

Finding a shorter side

If the hypotenuse is known, rearrange the formula and subtract the square of the known shorter side.

Example 2: A right-angled triangle has hypotenuse

13 cm

and one shorter side

5 cm

. Find the other shorter side.

Let the unknown side be $x$ .
$x^{2} + 5^{2} = 13^{2}$ .
$x^{2} + 25 = 169$ .
$x^{2} = 144$ .
$x = 12$ .

Checking whether a triangle is right-angled

Pythagoras can also be used backwards. Square all three side lengths and test whether the two smaller squares add to the largest square.

Example 3: Are side lengths

7 cm

24 cm

, and

25 cm

from a right-angled triangle?

Largest side is $25$ .
$7^{2} + 24^{2} = 49 + 576 = 625$ .
$25^{2} = 625$ .
The values match, so the triangle is right-angled.

Common mistakes

Using the formula on triangles that are not right-angled.
Choosing the wrong side as the hypotenuse.
Forgetting to square the side lengths before adding or subtracting.
Not taking the square root at the end.

Study routine

Mark the right angle first, label the hypotenuse clearly, then write the equation before substituting. In exams, a neat setup usually prevents most arithmetic errors.

Skills to practise

Pythagoras' theorem