Question

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

Explanation:

1. First, identify the coordinates of the two - points:

• Assume the two points are \((x_1,y_1)\) and \((x_2,y_2)\). From the graph, if one point is on the x - axis, say \((x_1,y_1)=(0,0)\) (the origin), and the other point has coordinates \((x_2,y_2)=(- 4,-2)\).

1. Then, use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\):

• Substitute \(x_1 = 0,y_1 = 0,x_2=-4,y_2 = - 2\) into the formula.
• \(d=\sqrt{(-4 - 0)^2+(-2 - 0)^2}=\sqrt{(-4)^2+(-2)^2}\).
• Calculate the squares: \((-4)^2=16\) and \((-2)^2 = 4\).
• So \(d=\sqrt{16 + 4}=\sqrt{20}\).

1. Simplify the radical:

• Factor 20: \(20=4\times5\).
• Then \(\sqrt{20}=\sqrt{4\times5}\).
• Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 4,b = 5\)), we get \(\sqrt{4}\cdot\sqrt{5}=2\sqrt{5}\).

Step1: Identify coordinates

Let the points be \((0,0)\) and \((-4,-2)\)

Step2: Apply distance formula

\(d=\sqrt{(-4 - 0)^2+(-2 - 0)^2}=\sqrt{16 + 4}\)

Step3: Simplify radical

\(d=\sqrt{4\times5}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)