Question

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

Explanation:

1. Identify the coordinates of the two - points:

• Assume the first point is \((x_1,y_1)=(2,1)\) and the second point is \((x_2,y_2)=(5, - 3)\) by observing the graph.

1. Apply the distance formula:

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Substitute \(x_1 = 2,y_1 = 1,x_2 = 5,y_2=-3\) into the formula:
• First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):
• \(x_2 - x_1=5 - 2 = 3\).
• \(y_2 - y_1=-3 - 1=-4\).
• Then, calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\):
• \((x_2 - x_1)^2+(y_2 - y_1)^2=3^2+( - 4)^2\).
• \(3^2+( - 4)^2 = 9 + 16=25\).
• Finally, find the distance \(d\):
• \(d=\sqrt{25}=5\).

Step1: Identify point coordinates

Let \((x_1,y_1)=(2,1)\) and \((x_2,y_2)=(5, - 3)\)

Step2: Apply distance formula

\(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{(5 - 2)^2+( - 3 - 1)^2}\)

Step3: Calculate differences

\(x_2 - x_1 = 3\), \(y_2 - y_1=-4\)

Step4: Calculate sum of squares

\((x_2 - x_1)^2+(y_2 - y_1)^2=3^2+( - 4)^2=9 + 16 = 25\)

Step5: Find distance

\(d=\sqrt{25}=5\)

Answer:

\(5\)