Question

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

Explanation:

Since the points on the graph are not clearly visible, assume the two - points are \((x_1,y_1)\) and \((x_2,y_2)\). The distance formula between two points in a coordinate plane is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step1: Identify the coordinates

Let the two points be \((x_1,y_1)\) and \((x_2,y_2)\).

Step2: Substitute into the formula

Calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\) first. Then take the square - root of the result: \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Simplify the expression under the square - root by expanding \((x_2 - x_1)^2=x_2^2-2x_1x_2 + x_1^2\) and \((y_2 - y_1)^2=y_2^2-2y_1y_2 + y_1^2\), and then combine like terms and simplify the square - root if possible.

Since we don't have the actual coordinates of the points, we can't give a numerical answer. But if the points are \((x_1,y_1)\) and \((x_2,y_2)\), the distance \(d\) between them in simplest radical form is given by the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

If we assume the points are \((3,4)\) and \((6,8)\) for example:

Step1: Identify \(x_1,y_1,x_2,y_2\)

Let \(x_1 = 3,y_1 = 4,x_2 = 6,y_2 = 8\).

Step2: Calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\)

\(x_2 - x_1=6 - 3=3\), so \((x_2 - x_1)^2 = 3^2=9\). \(y_2 - y_1=8 - 4 = 4\), so \((y_2 - y_1)^2=4^2 = 16\). Then \((x_2 - x_1)^2+(y_2 - y_1)^2=9 + 16=25\).

Step3: Find the square - root

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

In general, for two points \((x_1,y_1)\) and \((x_2,y_2)\), follow the above steps to get the distance in simplest radical form.

Answer:

\(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (actual value depends on the coordinates of the two points)