Question

calculate the distance between the points h=(0, 5) and k=(4, - 2) in the coordinate plane. give an exact answer (not a decimal approximation).

Explanation:

Step1: Identify distance formula

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

Step2: Assign values

Let $(x_1,y_1)=(0,5)$ and $(x_2,y_2)=(4, - 2)$. Then $x_2 - x_1=4 - 0 = 4$ and $y_2 - y_1=-2 - 5=-7$.

Step3: Calculate squares and sum

$(x_2 - x_1)^2=4^2 = 16$ and $(y_2 - y_1)^2=(-7)^2 = 49$. The sum is $16 + 49=65$.

Step4: Find distance

$d=\sqrt{65}$.

Answer:

$\sqrt{65}$