Question

find the distance between the points (6, 5) and (2, 1). write your answer as a whole number or a fully simplified radical expression. do not round.

Explanation:

Step1: Identify the distance - formula

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

Step2: Substitute the given points

Let $(x_1,y_1)=(6,5)$ and $(x_2,y_2)=(2,1)$. Then $x_2 - x_1=2 - 6=-4$ and $y_2 - y_1=1 - 5=-4$.

Step3: Calculate the squares and sum

$(x_2 - x_1)^2=(-4)^2 = 16$ and $(y_2 - y_1)^2=(-4)^2 = 16$. The sum is $16 + 16=32$.

Step4: Simplify the radical

$d=\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}$.

Answer:

$4\sqrt{2}$