NSW Year 8 - 2020 Edition

7.04 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

$a^2+b^2=c^2$`a`2+`b`2=`c`2,

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`=√`a`2+`b`2

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

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

- Look for right-angled triangles in the scenario
- Sketch a right-angled triangle showing all given information
- Choose which side, hypotenuse or a shorter side, you are trying to find
- 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.

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`?

Round your answer to two decimal places.

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.

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.

Find the value of $x$

`x`.Find the value of $y$

`y`.Round your answer to two decimal places.

applies Pythagoras' theorem to calculate side lengths in right-angled triangles, and solves related problems