- the diagonal and the two consecutive sides of a square
- the diagonal and the two consecutive sides of a rectangle
- the altitude and the two sides of an equilateral triangle
- the altitude and the two legs of an isosceles triangle

The **Pythagorean theorem** states that *for any right triangle the square of the length of its hypotenuse is equal to the sum of the squares of the lengths of its two legs*.

Mathematically, the theorem is expressed as follows

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

**a**and

**b**are the lengths of the legs and

**c**is the length of the hypotenuse of the right triangle.

If any two sides are known, then the third side can be calculated

a^{2} = c^{2} - b^{2}

b^{2} = c^{2} - a^{2}

In addition, the following properties also hold true for all right triangles:

- the hypotenuse is always the longest side and is always opposite the right angle
- the greater of the two acute angles is always opposite the longer leg

For a quick check of your calculations, refer to the **Pythagorean triples** given below. They are sets of three numbers that satisfy the Pythagorean theorem c^{2} = a^{2} + b^{2}. The table is arranged in ascending order by the hypotenuse.

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

3 | 4 | 5 |

6 | 8 | 10 |

5 | 12 | 13 |

9 | 12 | 15 |

8 | 15 | 17 |

12 | 16 | 20 |

7 | 24 | 25 |

15 | 20 | 25 |

24 | 7 | 25 |

10 | 24 | 26 |

20 | 21 | 29 |

18 | 24 | 30 |

16 | 30 | 34 |

21 | 28 | 35 |

12 | 35 | 37 |

15 | 36 | 39 |

24 | 32 | 40 |

9 | 40 | 41 |

27 | 36 | 45 |

14 | 48 | 50 |

30 | 40 | 50 |

24 | 45 | 51 |

20 | 48 | 52 |

28 | 45 | 53 |

33 | 44 | 55 |

40 | 42 | 58 |

36 | 48 | 60 |

60 | 11 | 61 |

56 | 33 | 65 |

60 | 25 | 65 |

60 | 32 | 68 |

56 | 42 | 70 |

55 | 48 | 73 |

60 | 45 | 75 |

When solving right triangles, there are two special right triangles that you might want to remember for their properties. Knowledge of their properties can simplify much of the calculations.

These are the:

- 45°-45°-90° right triangle
- 30°-60°-90° right triangle

An example of the 45°-45°-90° right triangle is given below.

Given isosceles right triangle ABC with two known and equal legs, find the length of its hypotenuse.

Applying the Pythagorean theorem, the hypotenuse can be calculated thus

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

c^{2} = 20^{2} + 20^{2}

c^{2} = 400 + 400

√c^{2} = √800

√c^{2} = √400 x √2

c = 20√2

Take note that the length of the hypotenuse is equal to the length of a leg multiplied by √2.The following problem deals with right triangles formed from a square.

Since any two consecutive sides of a square are equal, an isosceles 45°-45°-90° right triangle is also formed by the diagonal and two consecutive sides of a square as can be seen in the next diagram.

Given square ABCD, right triangle BCD is formed by BC and the two sides BD and CD. In this illustrated problem, the length of the diagonal

*d*is given while the length of the sides is unknown.

To get the length of the sides of square ABCD, the Pythagorean theorem will be applied.

With side BC as the hypotenuse and the sides BD and CD as the legs, we have

BD^{2} + CD^{2} = BC^{2}

Since BD = CD = s and BC = 10√2, the expression is reduced to an equation with one unknown.

s^{2} + s^{2} = (10√2)^{2}

2s^{2} = 100 x 2

s^{2} = 200/2

s^{2} = 100

√s^{2} = √100

s = 10

Observe that the length of the hypotenuse is equal to the length of a leg times √2.This gives us the general rule:

*For any 45°-45°-90° right triangle, the length of its hypotenuse is always equal to the product of its leg and √2*.

Another special right triangle that deserves attention is the 30°-60°-90° right triangle.

This kind of triangle is usually encountered in such a problem involving an equilateral triangle with unknown altitude (see the illustration below).

Here, triangle ABC is an equilateral triangle whose altitude

*h*, drawn from vertex C to the side AB, bisects vertex angle C and side AB thus forming two 30°-60°-90° right triangles.

In triangle ACD above, AC is the hypotenuse, CD and AD are the legs. Side CD, which is also the altitude of triangle ACD, is unknown.

To find the altitude, we use the Pythagorean theorem

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

AC^{2} = CD^{2} + AD^{2}

24^{2} = h^{2} + 12^{2}

h^{2} = 24^{2} - 12^{2}

h^{2} = 576 - 144

√h^{2} = √432

√h^{2} = √144 x √3

h = 12√3

The length of the altitude is one-half the length of the hypotenuse times √3.Let's take a look at another example of a 30°-60°-90° right triangle.

Given triangle ABC with c=28 and b=14, what is the altitude of the triangle?

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

28^{2} = a^{2} + 14^{2}

a^{2} = 28^{2} - 14^{2}

a^{2} = 784 - 196

√a^{2} = √588

√a^{2} = √196 x √3

a = 14√3

These examples confirm a general rule about a 30°-60°-90° right triangle: *its altitude is the product of one half the length of its hypotenuse and √3*.

For any other right triangles, the unknown side is calculated using the Pythagorean theorem as shown in the next three illustrated problems.

Given isosceles triangle ABC, find its altitude.

Its altitude CD bisects the base AB and forms two right triangles with its two congruent sides.

(For a discussion of the properties of an isosceles triangle and its altitude see Parts Of A Triangle And Their Graphs.)

Take note that the right triangles formed from this triangle are different from 30°-60°-90° right triangles formed from an equilateral triangle.

*Remember, an equilateral triangle is always an isosceles triangle but an isosceles triangle is not always an equilateral triangle.*

Applying the Pythagorean theorem in triangle ACD,

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

AC^{2} = CD^{2} + AD^{2}

15^{2} = h^{2} + 12^{2}

h^{2} = 15^{2} - 12^{2}

h^{2} = 225 - 144

√h^{2} = √81

h = 9

In rectangle ABCD shown below, the length of its diagonal BC is unknown.

Since diagonal BC forms a right triangle with sides BD and CD, we use the Pythagorean theorem to get the length of the diagonal.

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

BC^{2} = BD^{2} + CD^{2}

BC^{2} = 10^{2} + 24^{2}

BC^{2} = 100 + 576

√BC^{2} = √676

BC = 26

In this last problem, given rhombus ABCD with the lengths of its diagonals AD=12 and BC=16, find the length of its sides.

We know that the diagonals of a rhombus are perpendicular bisector of each other and that the two consecutive sides of a rhombus are equal.

Given these properties, we can take any of the four right triangles formed by the diagonals in order to obtain the length of one side of rhombus ABCD.

With CD as the hypotenuse of one of the right triangles, we find its length by applying the Pythagorean theorem

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

CD^{2} = 8^{2} + 6^{2}

CD^{2} = 64 + 36

√CD^{2} = √100

CD = 10

For a discussion of the properties of squares, rectangles rhombi and their diagonals, see the following topics: