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# Applications of the Pythagorean Theorem

## Height, distance, and angles or types of triangles.

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Applications of the Pythagorean Theorem

What if you were given an equilateral triangle in which all the sides measured 4 inches? How could you use the Pythagorean Theorem to find the triangle's altitude? After completing this Concept, you'll be able to solve problems like this one.

### Watch This

CK-12 Foundation: Applications of the Pythagorean Theorem

James Sousa: Pythagorean Theorem and Its Converse

### Guidance

##### Find the Height of an Isosceles Triangle

One way to use The Pythagorean Theorem is to find the height of an isosceles triangle (see Example A).

##### Prove the Distance Formula

Another application of the Pythagorean Theorem is the Distance Formula. We will prove it here.

Let’s start with point A(x1,y1)\begin{align*}A(x_1, y_1)\end{align*} and point B(x2,y2)\begin{align*}B(x_2, y_2)\end{align*}. We will call the distance between A\begin{align*}A\end{align*} and B,d\begin{align*}B, d\end{align*}.

Draw the vertical and horizontal lengths to make a right triangle.

Now that we have a right triangle, we can use the Pythagorean Theorem to find the hypotenuse, d\begin{align*}d\end{align*}.

d2d=(x1x2)2+(y1y2)2=(x1x2)2+(y1y2)2

Distance Formula: The distance between A(x1,y1)\begin{align*}A(x_1, y_1)\end{align*} and B(x2,y2)\begin{align*}B(x_2, y_2)\end{align*} is d=(x1x2)2+(y1y2)2\begin{align*}d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\end{align*}.

##### Classify a Triangle as Acute, Right, or Obtuse

We can extend the converse of the Pythagorean Theorem to determine if a triangle is an obtuse or acute triangle.

Acute Triangles: If the sum of the squares of the two shorter sides in a right triangle is greater than the square of the longest side, then the triangle is acute.

For b<c\begin{align*}b < c\end{align*} and a<c\begin{align*}a < c\end{align*}, if a2+b2>c2\begin{align*}a^2 + b^2 > c^2\end{align*}, then the triangle is acute.

Obtuse Triangles: If the sum of the squares of the two shorter sides in a right triangle is less than the square of the longest side, then the triangle is obtuse.

For b<c\begin{align*}b < c\end{align*} and a<c\begin{align*}a < c\end{align*}, if a2+b2<c2\begin{align*}a^2+b^2, then the triangle is obtuse.

#### Example A

What is the height of the isosceles triangle?

Draw the altitude from the vertex between the congruent sides, which will bisect the base.

72+h249+h2h2h=92=81=32=32=162=42

#### Example B

Find the distance between (1, 5) and (5, 2).

Make A(1,5)\begin{align*}A(1, 5)\end{align*} and B(5,2)\begin{align*}B(5, 2)\end{align*}. Plug into the distance formula.

d=(15)2+(52)2=(4)2+(3)2=16+9=25=5

Just like the lengths of the sides of a triangle, distances are always positive.

#### Example C

Graph A(4,1),B(3,8)\begin{align*}A(-4, 1), B(3, 8)\end{align*}, and C(9,6)\begin{align*}C(9, 6)\end{align*}. Determine if ABC\begin{align*}\triangle ABC\end{align*} is acute, obtuse, or right.

Use the distance formula to find the length of each side.

ABBCAC=(43)2+(18)2=49+49=98=(39)2+(86)2=36+4=40=(49)2+(16)2=169+25=194

Plug these lengths into the Pythagorean Theorem.

(98)2+(40)298+40138 ? (194)2 ? 194<194

ABC\begin{align*}\triangle ABC\end{align*} is an obtuse triangle.

CK-12 Foundation: Applications of the Pythagorean Theorem

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### Guided Practice

Determine if the following triangles are acute, right or obtuse.

1.

2.

3. A triangle with side lengths 5, 12, 13.

Set the longest side equal to c\begin{align*}c\end{align*}.

1.

62+(35)236+4581 ? 82 ? 64>64

The triangle is acute.

2.

152+142225+196421 ? 212 ? 441<441

The triangle is obtuse.

3. 52+122=132\begin{align*}5^2 +12^2 = 13^2\end{align*} so this triangle is right.

### Explore More

Find the height of each isosceles triangle below. Simplify all radicals.

Find the length between each pair of points.

1. (-1, 6) and (7, 2)
2. (10, -3) and (-12, -6)
3. (1, 3) and (-8, 16)
4. What are the length and width of a 42” HDTV? Round your answer to the nearest tenth.
5. Standard definition TVs have a length and width ratio of 4:3. What are the length and width of a 42” Standard definition TV? Round your answer to the nearest tenth.

Determine whether the following triangles are acute, right or obtuse.

1. 7, 8, 9
2. 14, 48, 50
3. 5, 12, 15
4. 13, 84, 85
5. 20, 20, 24
6. 35, 40, 51
7. 39, 80, 89
8. 20, 21, 38
9. 48, 55, 76

Graph each set of points and determine whether ABC\begin{align*}\triangle ABC\end{align*} is acute, right, or obtuse, using the distance formula.

1. A(3,5),B(5,8),C(2,7)\begin{align*}A(3, -5), B(-5, -8), C(-2, 7)\end{align*}
2. A(5,3),B(2,7),C(1,5)\begin{align*}A(5, 3), B(2, -7), C(-1, 5)\end{align*}
3. A(1,6),B(5,2),C(2,3)\begin{align*}A(1, 6) , B(5, 2), C(-2, 3)\end{align*}
4. A(6,1),B(4,5),C(5,2)\begin{align*}A(-6, 1), B(-4, -5), C(5, -2)\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.3.

### Vocabulary Language: English Spanish

acute triangle

acute triangle

A triangle where all angles are less than $90^\circ$.
Obtuse Triangle

Obtuse Triangle

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

Distance Formula

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

Vertex

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