# 4.27: The Pythagorean Theorem

- Page ID
- 4958

Square of the hypotenuse equals the sum of the squares of the legs in right triangles.

The two shorter sides of a **right triangle** (the sides that form the right angle) are the **legs** and the longer side (the side opposite the right angle) is the **hypotenuse**. For the Pythagorean Theorem, the legs are “\(a\)” and “\(b\)” and the hypotenuse is “\(c\)”.

**Pythagorean Theorem:** Given a right triangle with legs of lengths a and b and a hypotenuse of length \(c\), \(a^2+b^2=c^2\).

The converse of the Pythagorean Theorem is also true. It allows you to prove that a triangle is a right triangle even if you do not know its angle measures.

**Pythagorean Theorem Converse:** If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If \(a^2+b^2=c^2\), then \(\Delta ABC\) is a right triangle.

##### Pythagorean Triples

A combination of three numbers that makes the Pythagorean Theorem true is called a **Pythagorean triple**. Each set of numbers below is a Pythagorean triple.

\(3,4,5 \qquad 5,12,13\qquad 7,24,25\qquad 8,15,17\qquad 9,12,15\qquad 10,24,26\)

Any multiple of a Pythagorean triple is also considered a Pythagorean triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10, which is another triple. To see if a set of numbers makes a Pythagorean triple, plug them into the Pythagorean Theorem.

What if you were told that a triangle had side lengths of 5, 12, and 13? How could you determine if the triangle were a right one?

Example \(\PageIndex{1}\)

What is the diagonal of a rectangle with sides 10 and 16?

**Solution**

For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find \(d\).

\(\begin{align*} 10^2+16^2=d^2 \\ 100+256=d^2 \\ 356&=d^2 \\ d&=\sqrt{356}=2\sqrt{89}\approx 18.87 \end{align*}\)

Example \(\PageIndex{2}\)

Do 6, 7, and 8 make the sides of a right triangle?

**Solution**

Plug the three numbers into the Pythagorean Theorem. Remember that the largest length will always be the hypotenuse, c. If \(6^2+7^2=8^2\), then they are the sides of a right triangle.

\(\begin{align*} 6^2+7^2&=36+49=85& \\ 8^2&=64&\qquad 85 \neq 64,\: so\: the \:lengths \:are \:not \:the \:sides \:of \:a \:right \:triangle.\end{align*}\)

Example \(\PageIndex{3}\)

Find the length of the hypotenuse.

**Solution**

Use the Pythagorean Theorem. Set \(a=8\) and \(b=15\). Solve for \(c\).

\(\begin{align*} 8^2+152&=c^2 \\ 64+225&=c^2 \\ 289&=c^2 \qquad Take\: the \:square \:root \:of \:both \:sides. \\ 172&=c\end{align*}\)

Example \(\PageIndex{4}\)

Is 20, 21, 29 a Pythagorean triple?

**Solution**

If \(20^2+21^2=29^2\), then the set is a Pythagorean triple.

\(\begin{align*} 20^2+21^2&=400+441=841 \\ 29^2&=841 \end{align*}\)

Therefore, 20, 21, and 29 is a Pythagorean triple.

Example \(\PageIndex{5}\)

Determine if the triangles below are right triangles.

Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent c.

**Solution**

\(\begin{align*} a^2+b^2&=c^2 \\ 82+162 &\stackrel{?}{=}(8\sqrt{5})2 \\ 64+256 &\stackrel{?}{=}64\cdot 5 \\ 320&=320\qquad Yes\end{align*} \)

\(\begin{align*} a^2+b^2&=c^2 \\ 22^2+24^2&\stackrel{?}{=}262 \\ 484+576 &\stackrel{?}{=}676 \\ 1060 &\neq 676\qquad No\end{align*}\)

## Review

Find the length of the missing side. Simplify all radicals.

- If the
**legs of a right triangle**are 10 and 24, then the hypotenuse is __________. - If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
- If the sides of a square are 16, then the diagonal is ____________.
- If the sides of a square are 9, then the diagonal is _____________.

Determine if the following sets of numbers are Pythagorean Triples.

- 12, 35, 37
- 9, 17, 18
- 10, 15, 21
- 11, 60, 61
- 15, 20, 25
- 18, 73, 75

Determine if the following lengths make a right triangle.

- 7, 24, 25
- \(\sqrt{5},2\sqrt{10},3\sqrt{5}\)
- \(2\sqrt{3},\sqrt{6},8\)
- 15, 20, 25
- 20, 25, 30
- \(8\sqrt{3},6,2\sqrt{39}\)

## Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2.

## Vocabulary

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

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 |
A combination of three numbers that makes the Pythagorean Theorem true. |

Circle |
A circle is the set of all points at a specific distance from a given point in two dimensions. |

Conic |
Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas. |

degenerate conic |
A degenerate conic is a conic that does not have the usual properties of a conic section. Since some of the coefficients of the general conic equation are zero, the basic shape of the conic is merely a point, a line or a pair of intersecting lines. |

Ellipse |
Ellipses are conic sections that look like elongated circles. An ellipse represents all locations in two dimensions that are the same distance from two specified points called foci. |

hyperbola |
A hyperbola is a conic section formed when the cutting plane intersects both sides of the cone, resulting in two infinite “U”-shaped curves. |

Hypotenuse |
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle. |

Legs of a Right Triangle |
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle. |

Parabola |
A parabola is the characteristic shape of a quadratic function graph, resembling a "U". |

Pythagorean number triple |
A Pythagorean number triple is a set of three whole numbers a,b and c that satisfy the Pythagorean Theorem, \(a^2+b^2=c^2\). |

Right Triangle |
A right triangle is a triangle with one 90 degree angle. |

## Additional Resources

Interactive Element

Video: Using The Pythagorean Theorem Principles - Basic

Activities: Pythagorean Theorem and Pythagorean Triples Discussion Questions

Study Aids: Pythagorean Theorem Study Guide

Practice: The Pythagorean Theorem

Real World: Pythagoras TV