Question

question
find the distance between the two points in simplest radical form.
(7, -1) and (3, -3)

Explanation:

Step1: Recall the 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, \(x_1 = 7,y_1=-1,x_2 = 3,y_2=-3\).

Step2: Substitute the values into the formula

First, calculate \(x_2 - x_1\) and \(y_2 - y_1\):
\(x_2 - x_1=3 - 7=-4\), \(y_2 - y_1=-3-(-1)=-3 + 1=-2\).
Then, \((x_2 - x_1)^2+ (y_2 - y_1)^2=(-4)^2+(-2)^2\).
\((-4)^2+(-2)^2 = 16 + 4=20\).

Step3: Simplify the radical

\(d=\sqrt{20}=\sqrt{4\times5}\). Since \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}(a = 4,b = 5,a\geq0,b\geq0)\), \(\sqrt{4\times5}=\sqrt{4}\cdot\sqrt{5}=2\sqrt{5}\).

Answer:

\(2\sqrt{5}\)