3.04 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 as you would normally.

Examples

Example 1

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

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+36	Evaluate the squares
\displaystyle x^{2}	\displaystyle =	\displaystyle 117	Evaluate 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

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+256	Evaluate the squares
\displaystyle x^{2}	\displaystyle =	\displaystyle 400	Evaluate 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 400-9	Evaluate the squares
\displaystyle y^{2}	\displaystyle =	\displaystyle 391	Evaluate 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