# 3.04 Applications of Pythagoras' theorem

Lesson

We have looked at how to find the hypotenuse and the short side of a right-angled triangle. We will now look at how we can apply this theorem to everyday situations. Let's quickly recap Pythagoras' theorem.

Pythagoras' theorem

$a^2+b^2=c^2$a2+b2=c2,

Where:

• $c$c is the length of the hypotenuse, and
• $a$a and $b$b are the lengths of the two shorter sides

We can rearrange this equation to find formulas for each side length.

Rearranging Pythagoras' theorem

To find the hypotenuse:        $c=\sqrt{a^2+b^2}$c=a2+b2

To find a shorter side:     $a=\sqrt{c^2-b^2}$a=c2b2

To apply Pythagoras' theorem to real life situations, we can follow these four simple steps.

1. Look for right-angled triangles in the scenario
2. Sketch a right-angled triangle showing all given information
3. Choose which side, hypotenuse or a shorter side, you are trying to find
4. Substitute the known values in to the appropriate formula and solve as you would normally

Let's look at some examples so we can see this in action.

#### Practice questions

##### Question 1

The screen on a handheld device has dimensions $9$9 cm by $6$6 cm, and a diagonal of length $x$x cm.

What is the value of $x$x?

##### Question 2

The top of a flag pole is $4$4 metres above the ground and the shadow cast by the flag pole is $9$9 metres long.

The distance from the top of the flag pole to the end of its shadow is $d$d m. Find $d$d, rounded to two decimal places.

##### Question 3

A sports association wants to redesign the trophy they award to the player of the season. The front view of one particular design is shown below.

1. Find the value of $x$x.

2. Find the value of $y$y.