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 \\( \square \\).
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

Step1: Recall 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 part b, \(x_1 = 0,y_1=9,x_2 = 12,y_2 = 0\).

Step2: Substitute the values into the formula

Substitute into the formula: \(d=\sqrt{(12 - 0)^2+(0 - 9)^2}\).
First, calculate \((12 - 0)^2=12^2 = 144\) and \((0 - 9)^2=(-9)^2=81\).
Then \(d=\sqrt{144 + 81}\).

Step3: Simplify the expression

\(144+81 = 225\), so \(d=\sqrt{225}\).
Since \(\sqrt{225}=15\).

Answer:

15