Question

for each of the following pairs of points, find the length of \\( \overline { a b } \\).
a. \\( a ( 0,3 ) , b ( 0,2 ) \\)
b. \\( a ( 0,9 ) , b ( 12,0 ) \\)
c. \\( a ( - 21,12 ) , b ( - 25,11 ) \\)
a. the length of \\( \overline { a b } \\) is 1.
(type an exact answer, using radicals as needed. simplify your answer.)
b. the length of \\( \overline { a b } \\) is 15.
(type an exact answer, using radicals as needed. simplify your answer.)
c. the length of \\( \overline { a b } \\) is \\( \square \\).
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

Step1: Use the distance formula

The distance formula between two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(-21,12)\) and \(B(-25,11)\), \(x_1=-21\), \(y_1 = 12\), \(x_2=-25\), \(y_2 = 11\).

Step2: Substitute the values into the formula

First, find \(x_2 - x_1\) and \(y_2 - y_1\):
\(x_2 - x_1=-25-(-21)=-25 + 21=-4\)
\(y_2 - y_1=11 - 12=-1\)
Then, substitute into the distance formula:
\(d=\sqrt{(-4)^2+(-1)^2}=\sqrt{16 + 1}\)

Step3: Simplify the expression

\(\sqrt{16+1}=\sqrt{17}\)

Answer:

\(\sqrt{17}\)