7.06 Problem solving with the Pythagorean theorem

Remember that when working with the Pythagorean theorem, we must be working with a right triangle.

We can rearrange the Pythagorean theorem to find formulas for each side length.

Rearranging the Pythagorean theorem:

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,

1. Look for right triangles

2. Choose which side, hypotenuse or a shorter side, you are trying to find

3. Find the lengths of the other two sides

4. Apply the relevant formula and substitute the lengths of the other two sides

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

Examples

Example 1

Consider a cone with slant height 13\, \text{m} and perpendicular height 5\, \text{m}.

a

Find the length of the radius, r, of the base of this cone.

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2\,.

Apply the idea

We let b=r. Based on the given diagram, we are given a=5 and c=13.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 13^2	\displaystyle =	\displaystyle 5^2+r^2	Substitute the values
\displaystyle 169	\displaystyle =	\displaystyle 25 + r^2	Evaluate the squares
\displaystyle r^2	\displaystyle =	\displaystyle 169 - 25	Rearrange
	\displaystyle =	\displaystyle 144	Evaluate
\displaystyle \sqrt{r^2}	\displaystyle =	\displaystyle \sqrt{144}	Apply a square root to both sides
\displaystyle r	\displaystyle =	\displaystyle 12	Evaluate the square root

The radius is 12 m.

Watch question walkthrough

b

Find the length of the diameter of the cone's base.

Worked Solution

Create a strategy

Use the fact that the diameter is double the radius.

Apply the idea

\displaystyle \text{diameter}	\displaystyle =	\displaystyle 2\times 12	Multiply the radius by 2
	\displaystyle =	\displaystyle 24\text{ m}	Evaluate

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Example 2

The screen on a handheld device has dimensions 8 \text{ cm} by 4 \text{ cm}, and a diagonal of length x \text{ cm}.

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

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2\,.

Apply the idea

We let c=x. Based on the given diagram, we are given a=8 and b=4.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle x^2	\displaystyle =	\displaystyle 8^2+4^2	Substitute the values
\displaystyle x^2	\displaystyle =	\displaystyle 64 + 16	Evaluate the squares
\displaystyle x^2	\displaystyle =	\displaystyle 80	Add
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \sqrt{80}	Apply a square root to both sides
\displaystyle x	\displaystyle =	\displaystyle 8.94\, \text{cm}	Evaluate the square root and round