Can somebody me and explain how they got these answers ( 25 points ).

Step-by-step explanation:

use the pythagorean theorem

a^2 +b^2 = c^2  where c is the hypotenuse and a and b are the legs

1.  x^2 + (sqrt(7))^2 =(sqrt(19)^2

  x^2 +7 = 19

 subtract 7 from both sides

x^2 = 12

take the square root

x = sqrt(12)

x = sqrt(4) sqrt(3)

x = 2sqrt(3)

2.  x^2 + 12^2 = 13^2

x^2 +144=169

subtract 144 from each side

x^2 = 25

take the square root of each side

sqrt(x^2) = sqrt(25)

x = 5

3.  5^2 + 6^2 = x^2

25 + 36 = x^2

61 = x^2

take the square root of each side

x = sqrt(61)

4.  10^2 + 24^2 = x^2

100 +576 = x^2

676 = x^2

sqrt(676) = sqrt(x^2)

26 =x

5.  x^2 + x^2 = (8sqrt(2))^2

2x^2 = 8^2 *(sqrt(2))^2

2x^2 = 64*2

divide by 2

x^2 = 64

take the square root on each side

sqrt(x^2) =sqrt(64)

x = 8

6.  6^2 + x^2 = 15^2

36+x^2 = 225

x^2 = 189

sqrt(x^2) = sqrt(189)

x = sqrt(9*21)

x = 3sqrt(21)