Pythagoras review | Year 9

01 / LEARN

Understand the method.

THE CORE RULE

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

For a right triangle, $c$ is the hypotenuse: the side opposite the right angle.

Find the hypotenuse first

Look for the right-angle marker. The side opposite it is the hypotenuse, even when the triangle is rotated. It is always the longest side.

Pythagoras connects squared lengths: the two smaller square areas add to the largest. It does not say that the two shorter lengths add to the hypotenuse.

{leg}^{2} + {leg}^{2} = {hypotenuse}^{2}

Write the equation before choosing an operation

For a missing hypotenuse, add the two known squares. For a missing leg, subtract the known leg square from the hypotenuse square.

Take the positive square root because the answer is a length. Check its units and whether the hypotenuse is still the longest side.

c = \sqrt{a^{2} + b^{2}}, a = \sqrt{c^{2} - b^{2}}

Check a proposed right triangle

Square the longest length and compare it with the sum of the other two squares. Equality confirms a right triangle; inequality means it is not right-angled.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Find the hypotenuse

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

Worked solution

1
Substitute the two legs into Pythagoras.
$c^{2} = 6^{2} + 8^{2}$
The unknown is the hypotenuse, so its square is the sum of the two leg squares.
2
Evaluate, then take the positive square root.
$\begin{aligned} c^{2} & = 36 + 64 = 100 \\ c & = \sqrt{100} = 10 \end{aligned}$
The resulting length $10$ is greater than either leg, as expected.

Answer

$10 cm$ .

EXAMPLE 02

Find a missing leg

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

Worked solution

1
Place the hypotenuse on its own side of the equation.
$x^{2} + 8^{2} = 17^{2}$
The $17$ m side is the hypotenuse; $x$ is one of the two shorter sides.
2
Subtract the known leg square.
$\begin{aligned} x^{2} & = 17^{2} - 8^{2} \\ = 289 - 64 = 225 \end{aligned}$
Subtracting $8^{2}$ isolates the unknown square. Adding would find the wrong length.
3
Take the positive square root and check.
$x = \sqrt{225} = 15$
The check is $8^{2} + 15^{2} = 64 + 225 = 289 = 17^{2}$ .

Answer

$15 m$ .

03 / YOUR TURN

Try it yourself.

Calculate, explain, then apply.

CHECK 01

Find the missing length

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

Show answer & reasoning

$13$ .

$c^{2} = 5^{2} + 12^{2} = 25 + 144 = 169$ , so $c = 13$ . The hypotenuse is longer than both legs.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the method.

Find the hypotenuse first

Write the equation before choosing an operation

See the method in action.

Find the hypotenuse

Substitute the two legs into Pythagoras.

Evaluate, then take the positive square root.

Find a missing leg

Place the hypotenuse on its own side of the equation.

Subtract the known leg square.

Take the positive square root and check.

Try it yourself.

Find the missing length

Keep practising