Question

find the distance between the two points in simplest radical form.

Explanation:

Step1: Identify coordinates of points

First, we need to find the coordinates of the two yellow points. Let's assume the first point (let's say \( P_1 \)) is at \( (-7, -4) \) and the second point ( \( P_2 \)) is at \( (-3, -6) \). (We determine the coordinates by looking at the x - axis and y - axis values. For the x - coordinate, we move left/right from the origin, and for the y - coordinate, we move up/down. )

Step2: Apply distance formula

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

Substitute \( x_1=-7,y_1 = - 4,x_2=-3,y_2=-6 \) into the formula:

First, calculate \( x_2 - x_1=-3-(-7)=-3 + 7 = 4 \)

Then, calculate \( y_2 - y_1=-6-(-4)=-6 + 4=-2 \)

Now, substitute these values into the distance formula:

\( d=\sqrt{(4)^2+(-2)^2}=\sqrt{16 + 4}=\sqrt{20} \)

Step3: Simplify the radical

We can simplify \( \sqrt{20} \) by factoring 20. \( 20=4\times5 \), so \( \sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5} \)

Answer:

\( 2\sqrt{5} \)