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13.04 Applications of Pythagoras' theorem

Lesson

Introduction

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.

Applications of Pythagoras' theorem

Pythagoras theorem: a^{2}+b^{2}=c^{2}, where c is the length of the hypotenuse, and a and b are the lengths of the two shorter sides. We can rearrange this equation to find formulas for each side length.

To find the hypotenuse: c=\sqrt{a^2+b^2}. To find a shorter side: a=\sqrt{c^2-b^2}

To apply the Pythagorean 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.

Examples

Example 1

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

A cell phone screen with diagonal length of x centimetres, and side lengths of 6 centimetres and 9 centimetres.

Find the value of x, correct to two decimal places.

Worked Solution
Create a strategy

Use the Pythagoras' theorem: a^{2}+b^{2}=c^{2}

Apply the idea
\displaystyle x^{2}\displaystyle =\displaystyle 9^{2}+6^{2}Substitute a=9,\,b=6,\,and c=x
\displaystyle x^{2}\displaystyle =\displaystyle 81+36Evaluate the squares
\displaystyle x^{2}\displaystyle =\displaystyle 117Evaluate the sum
\displaystyle x\displaystyle =\displaystyle \sqrt{117}Take the square root of both sides
\displaystyle x\displaystyle =\displaystyle 10.82\text{ cm}Simplify and round to two decimal places

Example 2

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 in the diagram:

A trophy made from 2 right angled triangles joined at the hypotenuse. Ask your teacher for more information.
a

Find the value of x.

Worked Solution
Create a strategy

We can use the Pythagoras' theorem: a^{2}+b^{2}=c^{2}.

Apply the idea

We can apply Pythagoras' theorem to the bottom triangle that has side lengths of 16 cm and 12 cm, since the only unknown side is the hypotenuse.

\displaystyle x^{2}\displaystyle =\displaystyle 12^{2}+16^{2}Substitute a=12,\,b=16,\,and c=x
\displaystyle x^{2}\displaystyle =\displaystyle 144+256Evaluate the squares
\displaystyle x^{2}\displaystyle =\displaystyle 400Evaluate the sum
\displaystyle x\displaystyle =\displaystyle \sqrt{400}Take the square root of both sides
\displaystyle x\displaystyle =\displaystyle 20\text{ cm}Evaluate
b

Find the value of y, correct to two decimal places.

Worked Solution
Create a strategy

We can use the rearranged Pythagoras' theorem: a^{2}=c^{2}-b^{2}.

Apply the idea
\displaystyle y^{2}\displaystyle =\displaystyle 20^{2}-3^{2}Substitute a=y,\,b=3,\,and c=20
\displaystyle y^{2}\displaystyle =\displaystyle 400-9Evaluate the squares
\displaystyle y^{2}\displaystyle =\displaystyle 391Evaluate the difference
\displaystyle y\displaystyle =\displaystyle \sqrt{391}Take the square root of both sides
\displaystyle y\displaystyle =\displaystyle 19.77\text{ cm}Simplify and round to two decimal places
Idea summary

To apply the Pythagorean 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.

Outcomes

MA4-16MG

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

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