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### Pythagorean Theorem

- A
*right*triangle is a triangle which has a right angle. The side opposite the right angle, called the*hypotenuse*, is the longest side of the triangle. The two sides forming the right angle are called the*legs*of the triangle. - The Pythagorean Theorem states that if a right triangle has legs of lengths a and b, and hypotenuse of length c, then a
^{2}+ b^{2}= c^{2}. - This result can be used to find the length of any side of a right triangle if the other two sides are known.
- Some triples (a, b, c) of whole numbers, such as (3, 4, 5), satisfy the Pythagorean Theorem. Such triples are called
*Pythagorean triples*. Note that any multiple of a Pythagorean triple is also a Pythagorean triple. - The converse of the Pythagorean Theorem, which is also true, states that if the sides a, b, and c of a triangle satisfy the equation a
^{2}+ b^{2}= c^{2}, then the triangle is a right triangle.

### Pythagorean Theorem

The legs of a right triangle have lengths 5 and 12. Find the length of the hypothenuse.

- a
^{2}+ b^{2}= c^{2} - 5
^{2}+ 12^{2}= c^{2} - 25 + 144 = c
^{2} - 169 = c
^{2}

c = 13

The legs of a right triangle have lengths 10 and 8. Find the length of the hypothenuse.

- a
^{2}+ b^{2}= c^{2} - 8
^{2}+ 10^{2}= c^{2} - 64 + 100 = c
^{2} - 164 = c
^{2}

c = 2√{41}

The legs of a right triangle have lengths 16 and 22. Find the length of the hypothenuse.

- 16
^{2}+ 22^{2}= c^{2} - 256 + 484 = c
^{2} - 740 = c
^{2}

c = √{2 ×5 ×2 ×37} = 2√{185}

The hypothenuse of a right triangle has length 36 and the triangle is isoceles. Find the length of the legs.

- x
^{2}+ x^{2}= 36^{2} - 2x
^{2}= 1296 - x
^{2}= 648

x = √{9 ×9 ×2 ×2 ×2} = 18√2

The hypothenuse of a right triangle has length 12 and the triangle is isoceles. Find the length of the legs.

- x
^{2}+ x^{2}= 12^{2} - 2x
^{2}= 144 - x
^{2}= 72

x = √{2 ×2 ×2 ×3 ×3} = 6√2

The hypothenuse of a right triangle has length 16 and the triangle is isoceles. Find the length of the legs.

- x
^{2}+ x^{2}= 16^{2} - 2x
^{2}= 256 - x
^{2}= 128

x = √{4 ×4 ×2 ×2 ×2} = 8√2

The hypothenuse of a right triangle is 18 and the length of one leg is 6. Find the area of the triangle.

- 6
^{2}+ b^{2}= 18^{2} - 36 + b
^{2}= 324 - b
^{2}= 288 - b = √{2 ×2 ×2 ×2 ×2 ×3 ×3} = 12√2
- area = [1/2]bh
- area = [1/2]( 6 )( 12√2 )

area = 36√2

The hypothenuse of a right triangle is 9 and the length of one leg is 4. Find the area of the triangle.

- 4
^{2}+ b^{2}= 9^{2} - 16 + b
^{2}= 81 - b
^{2}= 65 - b = √{65}
- area = [1/2]( 4 )( √{65} )

area = 2√{65}

The longest side of a triangle has length 10 and the shortest side has length 7. Find the length of the third side so that the triangle is a right triangle.

- 10
^{2}= a^{2}+ 7^{2} - 100 = a
^{2}+ 49 - a
^{2}= 51

a = √{51}

The longest side of a triangle has length 30 and the shortest side has length 14. Find the length of the third side so that the triangle is a right triangle.

- 30
^{2}= a^{2}+ 14^{2} - 900 = a
^{2}+ 196 - a
^{2}= 704

a = √{8 ×8 ×11} = 8√{11}

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Pythagorean Theorem

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Right Triangles 0:06
- Vertex
- Hypotenuse
- Legs
- Pythagorean Theorem 1:21
- Graphical Representation
- Example
- Pythagorean Triples 3:40
- Example
- Converse of the Pythagorean Theorem 4:36
- Example
- Example 1: Length of Hypotenuse 7:24
- Example 2: Length of Legs 9:02
- Example 3: Area of Triangle 12:00
- Example 4: Length of Side 14:59

1 answer

Last reply by: Dr Carleen Eaton

Sun Apr 17, 2016 1:38 PM

Post by Oscar Prado on April 12, 2016

Why is the reason that we have to get the square root at the end of every problem?