# 4.36: Distance and Triangle Classification Using the Pythagorean Theorem

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Find missing sides to calculate the area.

## Applications of the Pythagorean Theorem

### 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 1).

#### Prove the **Distance Formula**

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

Let’s start with point \(A(x_1,y_1)\) and point \(B(x_2, y_2)\). We will call the distance between \(A\) and \(B\), \(d\).

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, \(d\).

\(\begin{align*} d^2=(x_1−x_2)^2+(y_1−y_2)^2 \\ d=\sqrt{(x_1−x_2)^2+(y_1−y_2)^2} \end{align*} \)

**Distance Formula:** The distance between \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is \(d=\sqrt{(x_1−x_2)^2+(y_1−y_2)^2}\).

#### 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 \(b<c\) and \(a<c\), if \(a^2+b^2>c^2\), 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 \(b<c\) and \(a<c\), if \(a^2+b^2<c^2\), then the triangle is obtuse.

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?

Example \(\PageIndex{1}\)

What is the height of the isosceles triangle?

**Solution**

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

\(\begin{align*} 7^2+h^2&=9^2 \\ 49+h^2&=81 \\ h^2&=32 \\ h&=\sqrt{32}=\sqrt{16\cdot 2}=4\sqrt{2}\end{align*}\)

Example \(\PageIndex{2}\)

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

**Solution**

Make \(A(1,5)\) and \(B(5,2)\). Plug into the distance formula.

\(\begin{align*} d &=\sqrt{(1−5)^2+(5−2)^2} \\ &=\sqrt{(−4)^2+(3)^2} \\ &=\sqrt{16+9}=\sqrt{25}=5\end{align*}\)

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

Example \(\PageIndex{3}\)

Graph \(A(−4,1)\), \(B(3,8)\), and \(C(9,6)\). Determine if \(\Delta{ABC}\) is acute, obtuse, or right.

**Solution**

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

\(\begin{align*} AB&=\sqrt{(−4−3)^2+(1−8)^2}=\sqrt{49+49}=\sqrt{98} \\ BC &=\sqrt{(3−9)^2+(8−6)^2}=\sqrt{36+4}=\sqrt{40} \\ AC &=\sqrt{(−4−9)^2+(1−6)^2}=\sqrt{169+25}=\sqrt{194}\end{align*}\)

Plug these lengths into the Pythagorean Theorem.

\(\begin{align*} \sqrt{98})^2+(\sqrt{40})^2 &? (\sqrt{194})^2 \\ 98+40 &? 194 \\ 138 &< 194\end{align*}\)

\(\Delta{ABC}\) is an **obtuse triangle**.

*For Examples 4 and 5, determine if the triangles are acute, right or obtuse.*

Example \(\PageIndex{4}\)

Set the longest side to c.

**Solution**

\(\begin{align*} 15^2+14^2 &? \: 21^2 \\ 225+196 &? \: 441 \\ 421 &< 441\end{align*}\)

The triangle is obtuse.

Example \(\PageIndex{5}\)

Set the longest side to \(c\).

**Solution**

A triangle with side lengths 5, 12, 13.

\(5^2+12^2=13^2\) so this triangle is right.

## Review

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

Find the length between each pair of points.

- \((-1, 6)\) and \((7, 2)\)
- \((10, -3)\) and \((-12, -6)\)
- \((1, 3)\) and \((-8, 16)\)
- What are the length and width of a 42” HDTV? Round your answer to the nearest tenth.
- 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.

- 7, 8, 9
- 14, 48, 50
- 5, 12, 15
- 13, 84, 85
- 20, 20, 24
- 35, 40, 51
- 39, 80, 89
- 20, 21, 38
- 48, 55, 76

Graph each set of points and determine whether \Delta{ABC} is acute, right, or obtuse, using the distance formula.

- \(A(3,−5), B(−5,−8), C(−2,7)\)
- \(A(5,3), B(2,−7), C(−1,5)\)
- \(A(1,6), B(5,2), C(−2,3)\)
- \(A(−6,1), B(−4,−5), C(5,−2)\)

## Review (Answers)

To see the Review answers, open this PDF file and look for section 8.3.

## Resources

## Vocabulary

Term | Definition |
---|---|

acute triangle |
A triangle where all angles are less than 90∘. |

Obtuse Triangle |
An obtuse triangle is a triangle with one angle that is greater than 90 degrees. |

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

## Additional Resources

Interactive Element

Video: The Pythagorean Theorem and The Converse of the Pythagorean Theorem

Activities: Applications of the Pythagorean Theorem Discussion Questions

Practice: Distance and Triangle Classification Using the Pythagorean Theorem