10-1 The Pythagorean Theorem

Quick Review

Given the lengths of two sides of a right triangle, you can use the Pythagorean Theorem to find the length of the third side. Given the lengths of all three sides of a triangle, you can determine whether it is a right triangle.

Example

What is the side length x in the triangle below?

a 2 + b 2 = c 2 Pythagorean Theorem 15 2 + x 2 = 39 2 Substitute 15 for  a , x  for  b , and 39 for c . 225 + x 2 = 1521 Simplify . x 2 = 1296 Subtract 225 from each side . x = 36 Find the principal square root of each side . table with 5 rows and 3 columns , row1 column 1 , eh squared , plus , b squared , column 2 equals , c squared , column 3 cap pythagoreancap theorem , row2 column 1 , 15 squared , plus , x squared , column 2 equals , 39 squared , column 3 cap substitute15for . eh comma x , for , b comma . and39forc . . , row3 column 1 , 225 plus , x squared , column 2 equals , 1521 , column 3 cap simplify , . , row4 column 1 , x squared , column 2 equals , 1296 , column 3 cap subtract225fromeachside . . , row5 column 1 , x , column 2 equals 36 , column 3 cap findtheprincipalsquarerootofeachside . . , end table

Exercises

Use the triangle below. Find the missing side length. If necessary, round to the nearest tenth.

6. a = 2.5 , b = 6 eh equals 2.5 comma b equals 6
7. a = 3.5 , b = 12 eh equals 3.5 comma b equals 12
8. a = 1.1 , b = 6 eh equals 1.1 comma b equals 6
9. a = 13 , c = 85 eh equals 13 comma c equals 85
10. a = 6 , c = 18.5 eh equals 6 comma c equals , 18.5
11. b = 2.4 , c = 2.5 b equals 2.4 comma c equals 2.5
12. b = 8.8 , c = 11 b equals 8.8 comma c equals 11
13. a = 1 , c = 2.6 eh equals 1 comma c equals 2.6

Determine whether the given lengths can be side lengths of a right triangle.

14. 4, 7.5, 8.5
15. 22, 120, 122
16. 8, 40, 41
17. 1.6, 3, 3.4
18. 6, 24, 25
19. 18, 52.5, 55.5
20. 1.2, 6, 6.1
21. 0.7, 2.3, 2.5
22. 1.3, 8.4, 8.5

10-2 Simplifying Radicals

Quick Review

A radical expression is simplified if the following statements are true.

• The radicand has no perfect-square factors other than 1.
• The radicand contains no fractions.
• No radicals appear in the denominator of a fraction.

Example

What is the simplified form of 3 x 2 ? fraction square root of 3 x end root , over square root of 2 end fraction . question mark

3 x 2 = 3 x 2 ⋅ 2 2 Multiply by  2 2 .   = 6 x 4 Multiply numerators and denominators .   = 6 x 2 Simplify . table with 3 rows and 3 columns , row1 column 1 , fraction square root of 3 x end root , over square root of 2 end fraction , column 2 equals . fraction square root of 3 x end root , over square root of 2 end fraction . dot , fraction square root of 2 , over square root of 2 end fraction , column 3 cap multiplyby . fraction square root of 2 , over square root of 2 end fraction , . , row2 column 1 , , column 2 equals . fraction square root of 6 x end root , over square root of 4 end fraction , column 3 cap multiplynumeratorsanddenominators . . , row3 column 1 , , column 2 equals , fraction square root of 6 x end root , over 2 end fraction , column 3 cap simplify , . , end table

Exercises

Simplify each radical expression.

23. 3 14 ⋅ ( − 2 21 ) 3 square root of 14 dot open negative 2 square root of 21 close
24. 8 ⋅ 1 4 6 square root of 8 dot , 1 fourth , square root of 6
25. 25 a 3 4 a square root of fraction 25 , eh cubed , over 4 eh end fraction end root
26. 8 s 18 s 3 fraction square root of 8 s end root , over square root of 18 , s cubed end root end fraction
27. − 2 7 x 2 ⋅ 1 3 28 x 3 negative 2 , square root of 7 , x squared end root , dot , 1 third . square root of 28 , x cubed end root
28. 6 5 t 2 ⋅ 15 t 2 6 , square root of 5 , t squared end root , dot . square root of 15 , t squared end root
29. Open-Ended Write three radical expressions that have 4 2 s 4 , square root of 2 s end root  as their simplified form. What do the three expressions have in common? Explain.
30. Geometry The width of a rectangle is s. Its length is 3s. How long is a diagonal of the rectangle? Express your answer in simplified radical form.

Table of Contents