Pythagoras review | Year 9

01 / LEARN

The idea, step by step

Why the squares matter

Pythagoras connects the areas of squares drawn on the three sides of a right triangle. The two smaller square areas add to the largest square area. It does not say that the two shorter lengths add to the longest length.

Find the right-angle marker first. The side opposite it is the hypotenuse; it is the longest side, even when the sketch is rotated. Call that side c. Let a and b denote the two shorter sides.

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

Set up before calculating

For a missing hypotenuse, add the two known squares and take the positive square root. For a missing shorter side, subtract the known shorter-side square from the hypotenuse square.

Taking a square root returns squared units to ordinary length units. Check that the final hypotenuse remains longer than either leg.

A common trap

Adding side lengths: 6 + 8 = 14, so the hypotenuse must be 14.

Instead: Add the squares, then take the square root: the hypotenuse is 10, not 14.

02 / WORKED EXAMPLES

See the thinking

Example 1

Find the longest side

A right triangle has legs 6 cm and 8 cm. Find its hypotenuse.

$c^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$ .
$c = \sqrt{100} = 10$ .

10 cm

Example 2

Find a shorter side

A right triangle has hypotenuse 17 m and one leg 8 m. Find its other leg.

$x^{2} + 8^{2} = 17^{2}$ .
$x^{2} = 289 - 64 = 225$ .
$x = 15$ .

15 m

Check your understanding

A right triangle has legs 5 and 12. Find its hypotenuse.

Show answer

$c = \sqrt{25 + 144} = 13$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the hypotenuse with legs 8 cm and 15 cm.
2 marks
Find the missing leg: hypotenuse 13 m, other leg 5 m.
2 marks
Find the hypotenuse with legs 7 mm and 24 mm.
2 marks
Find the hypotenuse with legs 10 m and 11 m. Round to one decimal place.
2 marks
Can sides 6 cm, 8 cm and 11 cm make a right triangle? Explain.
3 marks
A rectangular sign is 9 cm by 12 cm. Find its diagonal.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the hypotenuse with legs 8 cm and 15 cm.
2 marks
$c^{2} = 64 + 225 = 289$ .
$c = 17$ cm.
17 cm
Find the missing leg: hypotenuse 13 m, other leg 5 m.
2 marks
$x^{2} = 169 - 25 = 144$ .
$x = 12$ m.
12 m
Find the hypotenuse with legs 7 mm and 24 mm.
2 marks
$c^{2} = 49 + 576 = 625$ .
$c = 25$ mm.
25 mm
Find the hypotenuse with legs 10 m and 11 m. Round to one decimal place.
2 marks
$c = \sqrt{100 + 121} = \sqrt{221} \approx 14.9$ m.
14.9 m
Can sides 6 cm, 8 cm and 11 cm make a right triangle? Explain.
3 marks
$6^{2} + 8^{2} = 100$ but $11^{2} = 121$ .
The required equality is false.
No
A rectangular sign is 9 cm by 12 cm. Find its diagonal.
3 marks
$d = \sqrt{9^{2} + 12^{2}} = \sqrt{225} = 15$ cm.
15 cm

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Mathematics

Year 9 similarity, Pythagoras and right-triangle trigonometric ratios, with prerequisite revision. Non-right-triangle sine/cosine rules are not required.

Official curriculum source ↗

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