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.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

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 $c = 10$ .

Skill quiz: Identify the hypotenuse

Mark the right angle and find the longest side before substituting into Pythagoras.

Pythagoras' theorem

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 $x$ .
$x^{2} + 5^{2} = 13^{2}$ .
$x^{2} + 25 = 169$ .
$x^{2} = 144$ .
$x = 12$ .

Skill quiz: Find a missing shorter side

Subtract the known shorter-side square from the hypotenuse square, then take the positive square root.

Pythagoras' theorem

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.

Skill quiz: Test the three side lengths

Compare the sum of the two smaller squares with the largest square.

Pythagoras' theorem

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.

Skill quiz: Check the setup and square root

Identify the hypotenuse, square the lengths and remember to take the square root at the end.

Pythagoras' theorem

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.

Skill quiz: Apply Pythagoras step by step

Label the sides, write the equation and substitute the measurements before calculating.

Pythagoras' theorem