The Theorem of Pythagoras (Grade 9) — Mathematics Grades 8–11

Using the Theorem of Pythagoras to find an unknown side of a right-angled triangle.

Gr 9 Term 2 — The Theorem of PythagorasTerm 2

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What gets asked

Calculate the length of the hypotenuse of a right-angled triangle.
Calculate the length of a shorter side of a right-angled triangle.
Use Pythagoras to find a missing length inside a composite figure.
Use a missing length you found to calculate a perimeter or an area.

You must be able to

Identify the hypotenuse as the side opposite the right angle.
Apply a squared plus b squared equals c squared to find the hypotenuse.
Rearrange the theorem to find a shorter side instead of the hypotenuse.
Check first that the triangle in the figure actually has a right angle.

Traps that cost marks

Adding the two shorter sides instead of adding their squares. Square both shorter sides first, add the squares, then take the square root.
Adding the squares when the unknown side is actually a shorter side. For a shorter side, subtract the smaller square from the larger square before taking the root.
Finding the missing length but forgetting to use it in the perimeter or area asked for. The theorem is usually only the first step. Check what the question actually wants.

Worked example

A ladder leans against a wall. It touches the wall 8 m up, and its foot is 6 m from the wall. Calculate the length of the ladder.

The ladder is the hypotenuse: $c^{2} = a^{2} + b^{2}$ .
$c^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$ .
$c = 100 = 10$ m.

Find a missing side

In a right-angled triangle, two sides always tell you the third.

right-angled triangle: A triangle with one angle of exactly $9 0^{\circ}$ .
hypotenuse: The side opposite the right angle. It is always the longest side.
square: A number multiplied by itself. The square of $5$ is $25$ , written $5^{2}$ .
square root: The number that multiplies by itself to give you the one you started with. The square root of $25$ is $5$ .

In a right-angled triangle, the two shorter sides squared and added give the hypotenuse squared. Written with letters, $a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse.

Find the hypotenuse by adding. Square both short sides, add them, then take the square root. The square root at the end is the step most often forgotten.

Find a short side by subtracting. The hypotenuse squared minus the known short side squared gives the missing one squared. Then take the square root.

So the first question is always the same: is the side I am looking for the hypotenuse or not? Find the right angle, and the side opposite it is the hypotenuse, however the triangle is turned.

The theorem only works in a right-angled triangle. In a shape made of several triangles, use it only on one that has a right angle marked.

Rules to remember

$a^{2} + b^{2} = c^{2}$ , with $c$ the hypotenuse

Examples

A right-angled triangle has short sides of

8

cm and

15

cm. Find the hypotenuse.

Worked answer

Square each short side: $8^{2} = 64$ and $1 5^{2} = 225$ .
Add them: $64 + 225 = 289$ .
That is the hypotenuse squared, so take the square root.
$289 = 17$ .
The hypotenuse is 17 cm.

Answer: $17$ cm

Adding the sides themselves would give $23$ , which is not a length this triangle has.

A ladder

13

m long leans against a wall. Its foot is

5

m from the wall. How high up the wall does it reach?

Worked answer

The ladder is opposite the right angle, so the ladder is the hypotenuse.
The height is a short side, so subtract: $1 3^{2} - 5^{2}$ .
$169 - 25 = 144$ .
$144 = 12$ .
The ladder reaches 12 m up the wall.

Answer: $12$ m

The unknown here is a short side, so the squares are subtracted, not added.

A rectangular garden plot is

12

m long and

9

m wide. A path runs from corner to corner. Find the total length of the path and the two sides it joins.

Worked answer

The path is the hypotenuse of a right-angled triangle with sides $12$ m and $9$ m.
$1 2^{2} + 9^{2} = 144 + 81 = 225$ .
$225 = 15$ , so the path is 15 m.
Now add the two sides: $15 + 12 + 9 = 36$ .
The total length is 36 m.

Answer: $36$ m

Finding the missing length is only half the question; it still had to be used.

Traps

Adding the two short sides instead of adding their squares. Square first, then add, then take the square root: $8^{2} + 1 5^{2} = 289$ and $289 = 17$ .
Adding the squares when the missing side is a short one. If you are looking for a short side, subtract: $1 3^{2} - 5^{2} = 144$ .
Working out the missing length and then forgetting to use it in the perimeter or area asked for. Read the last line of the question again before you write your answer down.

Learn and practise “Find a missing side” in the app →

Grade 9 The Theorem of Pythagoras

What gets asked

You must be able to

Traps that cost marks

Worked example

Find a missing side

Rules to remember

Examples

Traps

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