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6. 2D Geometry
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Grade 8

6.03 Applications of the Pythagorean theorem

Lesson
Practice

Applications of the Pythagorean theorem

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

Remember the Pythagorean theorem is:

\displaystyle a^2+b^2=c^2
\bm{a, b}
lengths of legs of the right triangle
\bm{c}
length of the hypotenuse

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. Substitute the known side lengths in to the Pythagorean theorem

  4. Solve for the unknown length

The Pythagorean theorem can be applied to many contextual situations, including but not limited to: archirecture, construction, sailing, and space flight.

The image shows a sailboat has traveled from a beach, the distance creating a right triangle with side lengths 6 miles and 8 miles.

Imagine a sailor wants to know how far they've traveled from a straight starting point to their current position, taking into account they've moved at an angle. They sailed 6 miles east and then 8 miles north.

Using the Pythagorean theorem, the sailor can calculate the direct distance back to the starting point.

A shed. The roof have height of 9 feet and base of 12 feet. Ask your teacher for more information.

Let's say a construction worker needs to create a very steep roof for a shed. They know how tall they want the roof to be and how wide the base should be. If the roof needs to be 9 feet high (up from the base) and the base is 12 feet across, they can use the Pythagorean theorem to find out how long the roof will be from the top of the roof to either end of the base.

Examples

Example 1

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

A cone with a slant height of 13 meters, perpendicular height 5 meters and unknown radius r.
a

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

Worked Solution
Create a strategy

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

Apply the idea

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

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Use the Pythagorean theorem
\displaystyle 5^2+r^2\displaystyle =\displaystyle 13^2Substitute a=5,\,b=r,\, and c=13
\displaystyle 25 + r^2\displaystyle =\displaystyle 169Evaluate the exponents
\displaystyle r^2\displaystyle =\displaystyle 169 - 25Subtract 25 from both sides
\displaystyle =\displaystyle 144Evaluate the subtraction
\displaystyle \sqrt{r^2}\displaystyle =\displaystyle \sqrt{144}Take the square root of both sides
\displaystyle r\displaystyle =\displaystyle 12Evaluate the square root

The radius is 12\text{ m}.

Reflect and check

Since this is a right triangle where all three sides are positive integers, this represents a Pythagorean triple.

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\cdot 12Multiply the radius by 2
\displaystyle =\displaystyle 24\text{ m}Evaluate the multiplication

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}.

A handheld device with a length of 8 centimeters, a height of 4 centimeters and a diagonal labeled as x centimeters.

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

Worked Solution
Create a strategy

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

Apply the idea

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

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Use the Pythagorean theorem
\displaystyle 8^2+4^2\displaystyle =\displaystyle x^2Substitute a=8,\,b=4,\, and c=x
\displaystyle 64 + 16\displaystyle =\displaystyle x^2Evaluate the exponents
\displaystyle x^2\displaystyle =\displaystyle 80Reverse and evaluate the addition
\displaystyle \sqrt{x^2}\displaystyle =\displaystyle \sqrt{80}Take the square root of both sides
\displaystyle x\displaystyle =\displaystyle 8.94\, \text{cm}Evaluate the square root and round

Example 3

A ladder is leaning against a wall. The base of the ladder is 6 feet away from the wall, and the ladder is 7 feet long. How far up the wall does the ladder reach? Round your answer to two decimal places.

Worked Solution
Create a strategy

Sketch a diagram to help you identify the hypotenuse and the legs. Use the Pythagorean Theorem to find missing side.

Apply the idea

We have that a=5,\,c=7,\, and we need to find b.

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Use the Pythagorean theorem
\displaystyle 5^2+b^2\displaystyle =\displaystyle 7^2Substitute a=5 and c=7
\displaystyle 25 + b^2\displaystyle =\displaystyle 49Evaluate the exponents
\displaystyle b^2\displaystyle =\displaystyle 49 - 25Subtract 25 from both sides
\displaystyle =\displaystyle 24Evaluate the subtraction
\displaystyle \sqrt{b^2}\displaystyle =\displaystyle \sqrt{24}Take the square root of both sides
\displaystyle b\displaystyle \approx\displaystyle 4.9Evaluate the square root and round

The ladder will reach about 4.9 feet up on the wall.

Idea summary

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. Substitute the known side lengths in to the Pythagorean theorem

  4. Solve for the unknown length

Outcomes

8.MG.4

The student will apply the Pythagorean Theorem to solve problems involving right triangles, including those in context.

8.MG.4e

Apply the Pythagorean Theorem, and its converse, to solve problems involving right triangles in context.

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