Question

find the distance between the two points. simplify your answers, and write the exact answer in simplest radical form for irrational answers. (-1, -2) and (1, 6)

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 $(x_1,y_1)=(-1,-2)$ and $(x_2,y_2)=(1,6)$. Then $x_2 - x_1=1-(-1)=2$ and $y_2 - y_1=6 - (-2)=8$.

Step3: Calculate the squares and sum

$(x_2 - x_1)^2=2^2 = 4$ and $(y_2 - y_1)^2=8^2 = 64$. The sum is $4 + 64=68$.

Step4: Simplify the radical

$d=\sqrt{68}=\sqrt{4\times17}=2\sqrt{17}$.

Answer:

$2\sqrt{17}$