Question

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

Explanation:

1. First, assume the two - point formula:

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's assume the first point is \((x_1,y_1)\) and the second point is \((x_2,y_2)\). From the graph, if we assume the first point has coordinates \((-8,8)\) and the second point has coordinates \((-4,6)\).

1. Then, calculate the differences in \(x\) and \(y\) coordinates:

• Calculate \(x_2 - x_1\): \(x_2=-4\), \(x_1 = - 8\), so \(x_2 - x_1=-4-(-8)=-4 + 8=4\).
• Calculate \(y_2 - y_1\): \(y_2 = 6\), \(y_1 = 8\), so \(y_2 - y_1=6 - 8=-2\).

1. Next, substitute into the distance formula:

• Substitute \(x_2 - x_1 = 4\) and \(y_2 - y_1=-2\) into \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• \(d=\sqrt{4^2+(-2)^2}=\sqrt{16 + 4}\).
• Then \(d=\sqrt{20}\).

1. Finally, simplify the radical:

• We can factor 20 as \(20=4\times5\).
• So \(\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\).

Step1: Identify the distance formula

\(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step2: Find \(x_2 - x_1\)

\(x_2 - x_1=-4-(-8)=4\)

Step3: Find \(y_2 - y_1\)

\(y_2 - y_1=6 - 8=-2\)

Step4: Substitute into the formula

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

Step5: Simplify the radical

\(d = 2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)