Pythagoras | Year 11 | Ascent Library

01 / LEARN

Understand the idea.

Connect the three side lengths

For a right triangle, Pythagoras connects the two legs and hypotenuse: $c^{2} = a^{2} + b^{2}$ , where $c$ is opposite the right angle. Identifying $c$ comes before substituting numbers, regardless of which side is unknown.

The equation first gives a squared length. Take its positive square root because a physical length is positive. Keep an exact root during intermediate calculations and round only the final requested measurement.

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

Subtract when a leg is missing

When a leg is unknown, rearrange by subtraction: $a^{2} = c^{2} - b^{2}$ . Adding the two known squares would find a new hypotenuse instead and incorrectly make the missing leg longer than the given hypotenuse.

Translate the setting into a justified right triangle. A vertical wall and level ground meet at a right angle; the ladder is the sloping hypotenuse. A rectangle's diagonal likewise forms a triangle with two perpendicular sides.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

A right triangle has legs 9 m and 12 m. Find the hypotenuse.

Worked solution

1
Place the hypotenuse in its correct role.
$c^{2} = 9^{2} + 12^{2}$
The two supplied lengths are legs of a right triangle.
2
Add the squared lengths.
$c^{2} = 81 + 144 = 225$
Pythagoras adds the leg squares to obtain the hypotenuse square.
3
Take the positive square root.
$c = \sqrt{225} = 15 m$
A side length is positive, and fifteen exceeds both nine and twelve as a hypotenuse should.

Answer

15 m

EXAMPLE 02

Worked example 2

A 10 m ladder rests on level ground against a vertical wall. Its foot is 6 m from the wall. How high does it reach?

Worked solution

1
Set up the right-triangle equation.
$h^{2} + 6^{2} = 10^{2}$
The vertical wall and level ground form the legs; the ladder is the ten-metre hypotenuse.
2
Isolate the squared height.
$h^{2} = 100 - 36 = 64$
Subtract the ground-distance square from the hypotenuse square.
3
Take the positive square root.
$h = \sqrt{64} = 8 m$
The height must be positive and less than the ladder length.

Answer

8 m

03 / YOUR TURN

Try it yourself.

Check your understanding

A right triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.

Show answer & reasoning

15 cm

The missing leg is $\sqrt{17^{2} - 8^{2}} = \sqrt{289 - 64} = \sqrt{225} = 15 cm$ . The hypotenuse is seventeen centimetres, so its square must be the larger term.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
2 marks
A right triangle has hypotenuse 13 m and one leg 5 m. Find the other leg.
2 marks
Find the diagonal of a square with side 6 cm. Give an exact answer and then one decimal place.
3 marks
A rectangular field is 9 m by 12 m. How far is it between opposite corners?
2 marks
A 15 m cable reaches from the top of a vertical post to the ground 9 m away. Find the post’s height.
2 marks
A student finds a missing leg in a triangle with hypotenuse 10 and other leg 6 by calculating $\sqrt{100 + 36}$ . Explain and correct the error.
3 marks

Read the worked solutions

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

Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
2 marks
$c^{2} = 25 + 144 = 169$ .
$c = 13 cm$ .
13 cm
A right triangle has hypotenuse 13 m and one leg 5 m. Find the other leg.
2 marks
$b^{2} = 169 - 25 = 144$ .
$b = 12 m$ .
12 m
Find the diagonal of a square with side 6 cm. Give an exact answer and then one decimal place.
3 marks
$d^{2} = 6^{2} + 6^{2} = 72$ .
$d = \sqrt{72} = 6 \sqrt{2} \approx 8.5 cm$ .
$6 \sqrt{2} cm$ ; 8.5 cm
A rectangular field is 9 m by 12 m. How far is it between opposite corners?
2 marks
The diagonal is the hypotenuse.
$d = \sqrt{9^{2} + 12^{2}} = 15 m$ .
15 m
A 15 m cable reaches from the top of a vertical post to the ground 9 m away. Find the post’s height.
2 marks
$h^{2} = 15^{2} - 9^{2} = 144$ .
$h = 12 m$ .
12 m
A student finds a missing leg in a triangle with hypotenuse 10 and other leg 6 by calculating $\sqrt{100 + 36}$ . Explain and correct the error.
3 marks
The unknown leg cannot exceed the hypotenuse.
$b^{2} = 100 - 36 = 64$ , so $b = 8$ .
Subtract instead: the missing leg is 8 units.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.

Official curriculum source ↗

In this lesson

Understand the idea.

Connect the three side lengths

Subtract when a leg is missing

See the method in action.

Worked example 1

Place the hypotenuse in its correct role.

Add the squared lengths.

Take the positive square root.

Worked example 2

Set up the right-triangle equation.

Isolate the squared height.

Take the positive square root.

Try it yourself.

Check your understanding

Keep practising