Question

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

Explanation:

Step1: Identify the coordinates of the two points

From the graph, the two points are \((6,9)\) and \((9,6)\). Let \((x_1,y_1)=(6,9)\) and \((x_2,y_2)=(9,6)\).

Step2: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Substitute the values: \(d=\sqrt{(9 - 6)^2+(6 - 9)^2}\)

Step3: Simplify the expression inside the square - root

First, calculate \((9 - 6)=3\) and \((6 - 9)=-3\). Then \((9 - 6)^2+(6 - 9)^2=3^2+(-3)^2\).
Since \(3^2 = 9\) and \((-3)^2=9\), we have \(3^2+(-3)^2=9 + 9=18\).

Step4: Simplify the square root

\(d=\sqrt{18}\). We can rewrite \(18\) as \(9\times2\), so \(\sqrt{18}=\sqrt{9\times2}\).
Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 9\), \(b = 2\)), we get \(\sqrt{9\times2}=\sqrt{9}\cdot\sqrt{2}=3\sqrt{2}\).

Answer:

\(3\sqrt{2}\)