The Pythagorean Theorem and the Distance Formula

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Use this relationship, known as the Pythagorean Theorem, to find the length of a side of a right triangle.

The Pythagorean Theorem

Example 1

Find m in the triangle below, to the nearest tenth.

m 2 + n 2 = k 2 m 2 + 7.8 2 = 9.6 2 m 2 = 9.6 2 − 7.8 2 = 31.32 m = 31.32 ≈ 5.6 table with 4 rows and 2 columns , row1 column 1 , m squared , plus , n squared , column 2 equals , k squared , row2 column 1 , m squared , plus , 7.8 squared , column 2 equals , 9.6 squared , row3 column 1 , m squared , column 2 equals , 9.6 squared , minus , 7.8 squared , equals , 31.32 , row4 column 1 , m , column 2 equals , square root of 31.32 , almost equal to 5.6 , end table

To find the distance between two points on the coordinate plane, use the distance formula.

The distance d between any two points ( x 1 , y 1 ) open , x sub 1 , comma , y sub 1 , close and ( x 2 , y 2 ) open , x sub 2 , comma , y sub 2 , close is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d equals . square root of open . x sub 2 , minus , x sub 1 . close squared . plus . open . y sub 2 , minus , y sub 1 . close squared end root

Example 2

Find the distance between ( − 3 , 2 )  and  ( 6 , − 4 ) . open negative 3 comma 2 close , and , open 6 comma negative 4 close .

Thus, d is about 10.8 units.

Exercises

In each problem, a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. Find each missing length. Round your answer to the nearest tenth.

1. c if a = 6 eh equals 6 and b = 8 b equals 8
2. a if b = 12 b equals 12 and c = 13 c equals 13
3. b if a = 8 eh equals 8 and c = 17 c equals 17
4. c if a = 10 eh equals 10 and b = 3 b equals 3
5. a if b = 100 b equals 100 and c = 114 c equals 114
6. b if a = 12.0 eh equals , 12.0 and c = 30.1 c equals , 30.1

Find the distance between each pair of points, to the nearest tenth.

7. ( 0 , 0 ) , ( 4 , − 3 ) open 0 comma 0 close comma open 4 comma negative 3 close
8. ( − 5 , − 5 ) , ( 1 , 3 ) open negative 5 comma negative 5 close comma open 1 comma 3 close
9. ( − 1 , 0 ) , ( 4 , 12 ) open negative 1 comma 0 close comma open 4 comma 12 close
10. ( − 4 , 2 ) , ( 4 , − 2 ) negative 4 comma 2 close comma open 4 comma negative 2 close
11. (0, 15), (17, 0)
12. ( − 8 , 8 ) , negative 8 comma 8 close comma (8, 8)
13. ( − 1 , 1 ) , ( 1 , − 1 ) negative 1 comma 1 close comma open 1 comma negative 1 close
14. ( − 2 , 9 ) , negative 2 comma 9 close comma (0, 0)
15. ( − 5 , 3 ) , negative 5 comma 3 close comma (4, 3)
16. (2, 1), (3, 4)
17. ( 3 , − 2 ) , open 3 comma negative 2 close comma (3, 5)
18. (5, 4), ( − 3 , 1 ) negative 3 comma 1 close

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