**a² + b² = c²**

**
Basics
**Pythagorean Theorem: a² + b² = c²

**a = altitude = height**

b = base

c = hypotenuse

The Pythagorean Theorem applies to a right triangle (a triangle in which one of the angles measures 90°).

Questions involving the Pythagorean Theorem require the calculation of squares and square roots.

is the squaring command.

is the square root command.

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Formulas (provided by **Mathematics Formula Sheet**)

a² + b² = c²

Formulas (NOT provided by ** Mathematics Formula Sheet**)

c² = a² + b²

c = √c²

b² = c² – a²

b = √b²

a² = c² – b²

a = √a²

In a right triangle, a = 5 and b = 12. What is c?

Question

** Answer**c = 13

** Answer Process**c² = a² + b²

c² = 5² + 12²

c² = 25 + 144

c² = 169

c = √c²

c = √169

c = 13

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5 + 12 | 5²+12² | c² = a² + b² |

169 | c² | |

169 | √169 | c = √c² |

13 | Answer | |

## Pythagorean Theorem |

In a right triangle, a = 6 and c = 10. What is b?

Question

** Answer**b = 8

** Answer Process**b² = c² – a²

b² = 10² – 6²

b² = 100 – 36

b² = 64

b = √b²

b = √64

b = 8

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10 – 6 | 10²-6² | b² = c² – a² |

64 | b² | |

64 | √64 | b = √b² |

8 | Answer | |

## Pythagorean Theorem |

In a right triangle, b = 3 and c = 5. What is a?

Question

** Answer**a = 4

** Answer Process**a² = c² – b²

a² = 5² – 3²

a² = 25 – 9

a² = 16

a = √a²

a = √16

a = 4

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5 – 3 | 5²-3² | a² = c² – b² |

16 | a² | |

16 | √16 | a = √a² |

4 | Answer | |

## Pythagorean Theorem |

A right triangular billboard has a height of 6 feet and base of 8 feet.

Question

How long is its hypotenuse?

**Answer
**10 feet

**Answer Process**

c² = a² + b²

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = √c²

c = √100

c = 10

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6 + 8 | 6²+8² | c² = a² + b² |

100 | c² | |

100 | √100 | c = √c² |

10 | Answer | |

## Pythagorean Theorem |

A right triangular billboard has a height of 5 feet and hypotenuse of 13 feet.

Question

How long is its base?

**Answer
**12 feet

**Answer Process
**b² = c² – a²

b² = 13² – 5²

b² = 169 – 25

b² = 144

b = √b²

b = √144

b = 12

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13 – 5 | 13²-5² | b² = c² – a² |

144 | b² | |

144 | √144 | b = √b² |

12 | Answer | |

## Pythagorean Theorem |

A right triangular billboard has a base of 12 feet and hypotenuse of 20 feet.

Question

How high is it?

**Answer
**16 feet

**Answer Process
**a² = c² – b²

a² = 20² – 12²

a² = 400 – 144

a² = 256

a = √a²

a = √256

a = 16

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20 – 12 | 20²-12² | a² = c² – b² |

256 | a² | |

256 | √256 | a = √a² |

16 | Answer | |

## Pythagorean Theorem |

Question

A snail can crawl 4 feet along a wall and then turn 90° to the left and crawl 3 feet along another wall. If the snail took a straight shortcut from the start to the end of these two walls, how far would the snail crawl?

**Answer
**5 feet

**Answer Process
**c² = a² + b²

c² = 3² + 4²

c² = 9 + 16

c² = 25

c = √c²

c = √25

c = 5

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3 + 4 | 3²+4² | c² = a² + b² |

25 | c² | |

25 | √25 | c = √c² |

5 | Answer | |

## Pythagorean Theorem |

Question

A snail can crawl 4 feet along a wall and then turn 90° to the left and crawl 3 feet along another wall. If the snail took a straight shortcut from the start to the end of these two walls, how much shorter was the shortcut than sticking to the walls?

**Answer
**2 feet

**Answer Process
**c² = a² + b²

c² = 3² + 4²

c² = 9 + 16

c² = 25

c = √c²

c = √25

c = 5

Subtract shortcut distance from wall distance.

a + b – c = 3 + 4 – 5 = 2

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3 + 4 | 3²+4² | c² = a² + b² |

25 | c² | |

25 | √25 | c = √c² |

5 | c | |

3 + 4 – 5 | 3+4-5 | Subtract shortcut from walls. a + b – c |

2 | Answer | |

## Pythagorean Theorem |

**Practice – Questions**

1. In a right triangle, a = 10 and b = 24. What is c?

2. In a right triangle, a = 50 and c = 130. What is b?

3. In a right triangle, b = 30 and c = 50. What is a?

4.

A ladder is leaning against a wall. Its base is 4.4 meters from the wall. Its top is 3.3 meters up the wall. How long is the ladder?

5.

A ladder is leaning against a wall. The ladder is 11 feet long. Its top is 6.6 feet up the wall.

How long is its base?

6.

A ladder is leaning against a wall. The ladder is 25 feet long. Its base is 20 feet from the wall. How high is the ladder up the wall?

7.

A snake can slither 32 feet along a wall and then turn 90° to the left and slither 24 feet along another wall. If the snake took a straight shortcut from the start to the end of these two walls, how far would the snail slither?

8. With reference to Question 7, how much shorter was the shortcut than sticking to the walls?

9.

A sports fan can run 800 feet along the sideline of a playing field and then turn 90° to the left and run 600 feet along another sideline of a playing field. At halftime, the sports fan can run a shortcut straight across the playing field from the start to the end of the two sidelines. What is the distance of this shortcut?

10. With reference to Question 9, how much longer was sticking to the sidelines than taking the shortcut?

**
Practice – Answers
**1. c = 26

2. b = 120

3. a = 40

4. 5.5 meters

5. 8.8 feet

6. 15 feet

7. 40 feet

8. 16 feet

9. 1000 feet

10. 400 feet