Question

find the distance between the two points in simplest radical form.
(5, 1) and (-3, -3)

Explanation:

Step1: Recall distance formula

The distance $d$ between points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Step2: Substitute given points

Let $(x_1,y_1)=(5,1)$ and $(x_2,y_2)=(-3,-3)$.
$d=\sqrt{(-3-5)^2+(-3-1)^2}$

Step3: Calculate differences

Compute the values inside the squares:
$d=\sqrt{(-8)^2+(-4)^2}$

Step4: Evaluate squared terms

Calculate the squares:
$d=\sqrt{64+16}$

Step5: Sum and simplify radical

Add the terms and simplify the square root:
$d=\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}$

Answer:

$4\sqrt{5}$