**Today’s Question of the Day:**

How do you find the side lengths of a right triangle?

Let’s start by reviewing some terms about a right triangle. A right triangle has two sides (or legs) and a hypotenuse (the side opposite the right angle). In the right triangle ΔABC, the __ legs__ are sides a and b and the

__is side c.__

**hypotenuse**The __ Pythagorean Theorem__ is defined by the sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). Since the area of a square is defined by side

^{2}, the area of square on leg a is a

^{2}. Then, the Pythagorean Theorem is,

`a`^{2} + b^{2} = c^{2}.

We can now try to test our knowledge!

__Question 1__: Solve for c.

For this example, a = 3, b = 4, and we want to solve for variable c.

Using the Pythagorean Theorem, we plug in a, b, and c: `3`

^{2} + 4^{2} = c^{2}

Now, we simplify the left side of the equation.

`9 + 16 = c` |
Simplify 3^{2} and 4^{2}. |

`25 = c` |
Add 9 and 16. |

`√25 = √c` |
Square root both sides of the equation. |

`5 = c` |
Solve for c. |

So, for this right triangle, a = 3, b = 4, and __ c = 5__.

__Question 2__: Solve for b.

For this example, a = 5, c = 13, and we want to solve for variable b.

Using the Pythagorean Theorem, we plug in a, b, and c: `5`

^{2} + b^{2} = 13^{2}

Now, we simplify the left side of the equation.

`25 + b` |
Simplify 5^{2} and 13^{2}. |

`b` |
Subtract 25 from 169. |

`√b` |
Square root both sides of the equation. |

`b = 12` |
Solve for b. |

In this right triangle, a = 5, __ b = 12__, and c = 13.

***Side Learning***

The right triangles from Question 1 (3-4-5) and Question 2 (5-12-13) are **Common Pythagorean Triples**. Some of the most well-known triples are listed below.

a |
b |
c |
---|---|---|

3 | 4 | 5 |

5 | 12 | 13 |

8 | 15 | 17 |

7 | 24 | 25 |

9 | 40 | 41 |

***Bonus Question***

We have already solved the first two Common Pythagorean triples. Show that the Pythagorean Theorem works for the other three triples.