Question

find the distance between the points (9, 1) and (3, 5). write your answer as a whole number or a fully simplified radical expression. do not round.

Explanation:

Step1: Identify distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(9,1)$ and $(x_2,y_2)=(3,5)$.

Step2: Substitute values

Substitute $x_1 = 9,y_1 = 1,x_2 = 3,y_2 = 5$ into the formula:
$d=\sqrt{(3 - 9)^2+(5 - 1)^2}=\sqrt{(-6)^2+(4)^2}$.

Step3: Calculate squares

$(-6)^2=36$ and $(4)^2 = 16$, so $d=\sqrt{36 + 16}$.

Step4: Add values inside square - root

$36+16 = 52$, so $d=\sqrt{52}$.

Step5: Simplify the radical

Factor 52: $52=4\times13$, then $\sqrt{52}=\sqrt{4\times13}=2\sqrt{13}$.

Answer:

$2\sqrt{13}$ units