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# Pythagorean Triples

## Indirect measurement applications

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Practice Pythagorean Triples
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Use the Pythagorean Theorem

Have you ever had to use indirect measurement? Take a look at this dilemma.

The students in Mrs. Richardson’s class need to use the Pythagorean Theorem to measure so that they can paint. Now they are left with the question of which ladder to use. Mrs. Richardson has told them that the height of the shed is 23 feet.

They have the choice of two different ladders. One ladder is 20 feet long and one ladder is 25 feet long.

“If we choose the 20 foot ladder and it is about 4 feet from the shed, how high up will it go?” Aran asked taking out a piece of paper and a pencil.

“I don’t know. We also have to think about the 25 foot ladder. If it is also 4 feet from the shed, how high up will it reach?” Amy said. “We need the Pythagorean Theorem for this one,” Aran said.

Both students took out a piece of paper and a pencil and began to work.

Learning to use the Pythagorean Theorem in real – world problems is important because of its many uses. Pay attention to this Concept and you will be able to figure out how high each ladder will reach by the end of it.

### Guidance

Do you remember the Pythagorean Theorem? Take a look.

To begin, let’s look at the parts of a right triangle.

The legs are the two sides of the triangle that are labeled \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. The hypotenuse is the longest side of a right triangle and it is labeled \begin{align*}c\end{align*}. There is a special relationship between the legs of a right triangle and the hypotenuse of a right triangle.

One of the special characteristics of right triangles is described by the Pythagorean Theorem, thought to have been developed around 500 B.C.E. It states that the squared value of the hypotenuse will equal the sum of the squares of the two legs. In the triangle above, the sum of the squares of the legs is \begin{align*}a^2 + b^2\end{align*} and the square of the hypotenuse is \begin{align*}c^2\end{align*}. So, the Pythagorean theorem is commonly represented as \begin{align*}a^2 + b^2 = c^2\end{align*} where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are the legs of the right triangle and \begin{align*}c\end{align*} is the hypotenuse.

The Pythagorean Theorem is known as \begin{align*}a^2 + b^2 = c^2\end{align*}.

Now that you are familiar with the Pythagorean Theorem, there are many different ways you can use it. Math questions will often require you to use the theorem even if it isn’t mentioned specifically in the text. Whenever you look at a shape, think about whether or not a right angle is present. If there is a right triangle in the shape, you may need to use the Pythagorean Theorem.

Write this statement down in your notebook.

There are many situations in which you may need to use the Pythagorean Theorem on objects that don’t even appear (at first) to be right triangles. Often, you will find that measurement involves a figure that looks like a right triangle. You can use the Pythagorean Theorem in these situations. This skill is called indirect measurement, and it is important to be comfortable using it in many situations.

Indirect measurement allows you to figure out lengths or distances that would be difficult by using logic and mathematical knowledge.

The situation below requires indirect measurement to solve the problem.

The diagram below shows the distance on roads to get from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*}. If you bike, however, you can travel in a straight line between those two points. What is the shortest possible distance between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}?

If you look at the image above, you see a rectangle, not a triangle. So, at first, you are unlikely to notice that you must use the Pythagorean Theorem to solve the problem. The question asks for the shortest distance between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}. You know from your geometry studies that the shortest distance between two points is always a straight line, so this distance will be a diagonal in the rectangle.

Now that the diagonal is drawn in, the triangle is more noticeable. In fact, this triangle is a 3:4:5 triangle, so you can quickly see that the hypotenuse will be 5 miles. Use the Pythagorean Theorem to confirm your answer.

Five miles is the correct answer. If you bike between points \begin{align*}A\end{align*} and \begin{align*}B\end{align*} on the map, and you can go in a straight line, the distance will be 5 miles.

Working with the Pythagorean Theorem in this way requires you to be a detective of sorts. When you see a figure, you can think about what the characteristics of the figure are. This can help you. A big clue in the last example is that a rectangle has \begin{align*}90^\circ\end{align*} angles. When you see a figure with \begin{align*}90^\circ\end{align*} angles, you will know that you can form right triangles in that figure.

Find each missing measure using the Pythagorean Theorem.

#### Example A

\begin{align*}1.5, 2, ?\end{align*}

Solution:  \begin{align*}2.5\end{align*}

#### Example B

\begin{align*}12, ?, 20\end{align*}

Solution:  \begin{align*}16\end{align*}

#### Example C

\begin{align*}18,24,?\end{align*}

Solution:  \begin{align*}30\end{align*}

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

First, let’s start with the ladder that is 20 feet long. We use the ladder length as the \begin{align*}c\end{align*} in the Pythagorean Theorem. It is the long side of the triangle. The height that the ladder will reach on the shed is what we are looking to solve, we will call that \begin{align*}a\end{align*} and that is our unknown value. We know that the ladder is four feet from the shed, and so this is our \begin{align*}b\end{align*} value.

Now we can look at the ladder that is 25 feet long.

If the shed is 23 feet high, then the students should use the ladder that is 25 feet long so that they can paint all the way to the top of the shed.

### Vocabulary

Indirect Measurement
using geometric properties to figure out distances and lengths that would otherwise be challenging to measure.
Pythagorean Theorem
the formula for figuring out the side lengths of a right triangle - \begin{align*}a^2+b^2=c^2\end{align*}
Pythagorean triple
different forms of the ratio 3:4:5 which represent the side lengths of a right triangle.

### Guided Practice

Here is one for you to try on your own.

Karen's yard is a square shape with 90 degree angles in each corner. She put a diagonal path from one end of the yard to the other end. If the diagonal is 35 feet long, how long is each side of the yard?

Solution

To figure this out, we can use the Pythagorean Theorem. We know that the diagonal is 35 feet long. Therefore side a and b need to correspond with the values needed to form a Pythagorean Triple.

If we use a 3,4,5 triangle as our model, we know that the hypotenuse is 35 feet long.

\begin{align*}5 \times 7 = 35\end{align*}

If we multiply the other two values in the model by 7, we should have the correct lengths.

\begin{align*}3 \times 7 = 21\end{align*}

Side a is 21 feet long.

\begin{align*}4 \times 7 = 28\end{align*}

Side b is 28 feet long.

### Practice

Directions: Identify whether or not each set of measurements indicates a Pythagorean Triple.

1. 3, 4, 5
2. 6, 8, 12
3. 6, 8, 10
4. 15, 20, 25
5. 5, 9, 14
6. 9, 12, 15
7. 18, 24, 30
8. 1.5, 2, 4
9. 1.5, 2, 2.5
10. 21, 28, 35

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 = 6, b = 10, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}a = 5, b = 7, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}a = 7, b = 9, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}a = 6, b = 8, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}a = 9, b = 12, c = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

### Vocabulary Language: English

Indirect Measurement

Indirect Measurement

Indirect measurement is the process of using the characteristics of similar triangles to measure distances.
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, $a^2 + b^2 = c^2$.