Question

find the distance between the two points in simplest radical form. (0, 3) and (6, 0)

Explanation:

Step1: Recall distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \(x_1 = 0,y_1 = 3,x_2=6,y_2 = 0\).

Step2: Substitute values into formula

Substitute the values into the formula:
\(d=\sqrt{(6 - 0)^2+(0 - 3)^2}\)
\(=\sqrt{6^2+(- 3)^2}\)
\(=\sqrt{36 + 9}\)

Step3: Simplify the expression

Simplify the expression inside the square root:
\(\sqrt{36+9}=\sqrt{45}\)
Factor \(45\) as \(9\times5\), so \(\sqrt{45}=\sqrt{9\times5}\)
Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we get \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\)

Answer:

\(3\sqrt{5}\)