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Applications of the Pythagorean Theorem

Height, distance, and angles or types of triangles.

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Practice Applications of the Pythagorean Theorem
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The Pythagorean Theorem, Perimeter and Area

Have you ever tried to measure a poster board? Take a look at this dilemma.

Lena had a piece of poster board that measured 12 inches by 16 inches. She cuts the poster board in half diagonally and wants to know the perimeter of one piece. What is the perimeter of half of Lena’s board?

Pay attention and you will know how to solve this problem by the end of the Concept.

Guidance

Do you remember how to identify perimeter and area? Do you know how to use the Pythagorean Theorem to calculate the perimeter and area of a figure.

Take a look.

The perimeter of an object is the distance around the outside of it.

So, for a triangle, it would be the sum of the two legs and the hypotenuse.

The area of an object is the amount of space it occupies, or how many square it would cover on a grid.

The area formula for a triangle is \begin{align*}A = \frac{1}{2} bh\end{align*} where \begin{align*}b\end{align*} is the length of the base of the triangle, and \begin{align*}h\end{align*} is the height. In a right triangle, the product \begin{align*}bh\end{align*} can always be found by multiplying the two legs together.

Sometimes, you won’t know all of the dimensions of a right triangle. In order to find the perimeter of a right triangle, you have to know all three sides. In order to find the area of a right triangle, you have to know the base and the height. If you don’t know all of the dimensions that you need to solve a problem, you can use the Pythagorean Theorem to help you in problem solving.

What is the perimeter and area of the triangle below?

The first thing to notice is that there is a missing leg of this triangle. Before we can do anything, we have to figure out the length of the missing side. This is a right triangle, so the first step in completing this problem is using the Pythagorean Theorem to identify the length of the missing leg.

\begin{align*}a^2 + b^2 = c^2\end{align*} , where \begin{align*}a = 5\end{align*} and \begin{align*}c = 13\end{align*}

\begin{align*}5^2 + b^2 & = 13^2\\ 25 + b^2 & = 169\\ 25 + b^2 - 25 & = 169 - 25\\ b^2 & = 144\\ \sqrt{b^2} & = \sqrt{144}\\ b & = 12\end{align*}

The length of the missing side is 12 inches.

Now, to find the perimeter \begin{align*}(P)\end{align*} of the triangle, add up the lengths of the three sides.

\begin{align*}P & = leg + leg + hypotenuse\\ P & = 5 + 12 +13\\ P & = 30\end{align*}

The perimeter of the triangle is 30 inches.

To find the area, use the area formula shown above. Remember that in this triangle, \begin{align*}b=5\end{align*} and \begin{align*}h=12\end{align*} . The height is the other leg because it is a right triangle.

\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2} (5)(12)\\ A & =\frac{1}{2} (60)\\ A & = 30\end{align*}

The area of the triangle is 30 square inches. Remember to use square units when measuring area.

This triangle is unique in that the numerical value for both the perimeter and the area are the same. Remember however, that the true values are different because the units are different.

Example A

What is the perimeter of a triangle with side lengths of 6, 8 and 10?

Solution: 24

Example B

If a triangle has leg lengths of 12 and 16. What is the length of the hypotenuse?

Solution: 20

Example C

What is the area and the perimeter of the triangle in Example B?

Solution: Area = 96, Perimeter = 48

Now let's go back to the dilemma from the beginning of the Concept.

To find the perimeter, you must find the length of the missing side (the hypotenuse). You may recognize 12 and 16 as being multiples of 3 and 4 (greater by a factor of 4) and could conclude that the hypotenuse will be four times five, or 20. However, you can solve this using the Pythagorean Theorem even if you don’t notice the Pythagorean triple.

\begin{align*}a^2 + b^2 = c^2\end{align*} , where \begin{align*}a = 12\end{align*} and \begin{align*}b = 16\end{align*}

\begin{align*}12^2 + 16^2 & = c^2\\ 144 + 256 & = c^2\\ 400 & = c^2\\ \sqrt{400} & = \sqrt{c^2}\\ 20 & = c\end{align*}

The missing side is 20 inches.

Don’t stop here, though! You have to find the perimeter of the triangle, so add the three sides together.

\begin{align*}P & = leg + leg + hypotenuse\\ P & = 12 + 16 + 20\\ P & = 48\end{align*}

The perimeter of half of Lena’s poster board is 48 inches.

Guided Practice

Here is one for you to try on your own.

What is the area of a right triangle that has one leg of 6 yards and a hypotenuse of 10 yards?

Solution

To figure this out, we must first use the Pythagorean Theorem to find the length of the missing leg.

\begin{align*}a^2 + b^2 = c^2\end{align*}

Now substitute in the given values and solve.

\begin{align*}6^2 + b^2 &= 10^2 \\ 36 + b^2 &= 100 \\ b^2 &= 100 - 36 \\ b^2 &= 64\end{align*}

Finally we take the square root of 64. This is our answer.

\begin{align*}b = 8\end{align*}

Now we can find the area of the triangle.

\begin{align*}A &= \frac{1}{2}bh \\ A &= \frac{1}{2}(8)(6) \\ A &= \frac{1}{2}48 \\ A &= 24\end{align*}

The area of the triangle is 24 units.

Explore More

Directions: Find the missing side length of each right triangle by using the Pythagorean Theorem. You may round to the nearest tenth when necessary.

1. \begin{align*}a = 10, b = 14, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}a = 6, b = \underline{\;\;\;\;\;\;\;\;\;\;}, c = 10\end{align*}
3. \begin{align*}a = \underline{\;\;\;\;\;\;\;\;\;\;}, b = 12, c = 15\end{align*}
4. \begin{align*}a = 15, b = \underline{\;\;\;\;\;\;\;\;\;\;}, c = 25\end{align*}
5. \begin{align*}a = \underline{\;\;\;\;\;\;\;\;\;\;}, b = 32, c = 40\end{align*}
6. \begin{align*}a = 30, b = 40, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}a = 1.5, b = 2, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}a = 4.5, b = 6, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}a = 6.6, b = 8.8, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}a = 36, b = 48, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}a = 27, b = 36, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

1. A television is measured by the length of the diagonal from one corner to another. If the screen is 8 inches by 15 inches, what is the length of the diagonal?
2. Do the numbers 15, 20 and 25 comprise a Pythagorean triple?
3. What is the perimeter of a right triangle with a hypotenuse of 30 inches and a leg of 18 inches?
4. What is the area of a right triangle with a hypotenuse of 30 inches and a leg of 18 inches?

Vocabulary Language: English

Area

Area

Area is the space within the perimeter of a two-dimensional figure.
Distance Formula

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be defined as $d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
Indirect Measurement

Indirect Measurement

Indirect measurement is the process of using the characteristics of similar triangles to measure distances.
Obtuse Triangle

Obtuse Triangle

An obtuse triangle is a triangle with one angle that is greater than 90 degrees.
Perimeter

Perimeter

Perimeter is the distance around a two-dimensional figure.
Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.
Pythagorean Triple

Pythagorean Triple

A Pythagorean Triple is a set of three whole numbers $a,b$ and $c$ that satisfy the Pythagorean Theorem, $a^2 + b^2 = c^2$.
Vertex

Vertex

A vertex is a point of intersection of the lines or rays that form an angle.