Question

find the distance between the two points (-3,2) and (3,5).

Explanation:

Step1: Recall distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(-3,2)\) and \((x_2,y_2)=(3,5)\).

Step2: Substitute values into formula

Substitute \(x_1=-3\), \(y_1 = 2\), \(x_2=3\), \(y_2 = 5\) into the formula:
\(d=\sqrt{(3 - (-3))^2+(5 - 2)^2}=\sqrt{(3 + 3)^2+(3)^2}=\sqrt{6^2+3^2}\)

Step3: Simplify the expression

Calculate \(6^2 = 36\) and \(3^2=9\), then \(36 + 9=45\). So \(d=\sqrt{45}\), which simplifies to \(3\sqrt{5}\) (since \(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\)).

Answer:

The distance between the two points is \(3\sqrt{5}\) (or approximately \(6.71\) if a decimal approximation is preferred).