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 the first point be \((2,1)\) and the second point be \((4, - 3)\).

1. Then, identify the values of \(x_1,x_2,y_1,y_2\):

• Here, \(x_1 = 2,y_1 = 1,x_2 = 4,y_2=-3\).

1. Next, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):

• \(x_2 - x_1=4 - 2=2\).
• \(y_2 - y_1=-3 - 1=-4\).

1. Then, calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\):

• \((x_2 - x_1)^2+(y_2 - y_1)^2=2^2+( - 4)^2\).
• \(2^2+( - 4)^2 = 4 + 16=20\).

1. Finally, find the distance \(d\):

• \(d=\sqrt{20}\).
• Simplify \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\).

Step1: Identify distance formula

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

Step2: Determine point - coordinates

Let \((x_1,y_1)=(2,1)\) and \((x_2,y_2)=(4, - 3)\)

Step3: Calculate \(x_2 - x_1\) and \(y_2 - y_1\)

\(x_2 - x_1 = 2\), \(y_2 - y_1=-4\)

Step4: Calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\)

\((x_2 - x_1)^2+(y_2 - y_1)^2=4 + 16 = 20\)

Step5: Simplify the square - root

\(d=\sqrt{20}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)