Question

1. the endpoints of $overline{ab}$ are $a(3, - 4)$ and $b(7,2)$. determine and state the length of $overline{ab}$ in simplest radical form.

Explanation:

Step1: Identify 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, \(x_1 = 3,y_1=-4,x_2 = 7,y_2 = 2\).

Step2: Substitute the values into the formula

First, find \((x_2 - x_1)\) and \((y_2 - y_1)\):
\(x_2 - x_1=7 - 3=4\)
\(y_2 - y_1=2-(-4)=2 + 4=6\)
Then, find \((x_2 - x_1)^2+(y_2 - y_1)^2\):
\((x_2 - x_1)^2+(y_2 - y_1)^2=4^2+6^2=16 + 36=52\)

Step3: Simplify the radical

\(d=\sqrt{52}=\sqrt{4\times13}=\sqrt{4}\times\sqrt{13}=2\sqrt{13}\)

Answer:

The length of \(\overline{AB}\) is \(2\sqrt{13}\)