Question

1. find the distance between the two points in simplest radical form. (-4, -6) and (-1, -3)

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, \(x_1=-4,y_1 = - 6,x_2=-1,y_2=-3\).

Step2: Substitute values into formula

Substitute the values into the formula:
\(d=\sqrt{(-1 - (-4))^2+(-3 - (-6))^2}\)
Simplify the expressions inside the square root:
\(-1-(-4)=-1 + 4=3\)
\(-3-(-6)=-3 + 6 = 3\)
So, \(d=\sqrt{(3)^2+(3)^2}\)

Step3: Simplify the square root

Calculate the squares:
\((3)^2 = 9\), \((3)^2=9\)
Then, \(d=\sqrt{9 + 9}=\sqrt{18}\)
Simplify \(\sqrt{18}\) by factoring 18 as \(9\times2\):
\(\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\)

Answer:

\(3\sqrt{2}\)