Question

are the endpoints of the segments \\(\overline{ab}\\), \\(\overline{cd}\\), and \\(\overline{ef}\\). \\(a(-1, 6)\\), \\(b(4, 8)\\) \\(c(0, -3)\\), \\(d(-2, -8)\\) \\(e(-5, 3)\\), \\(f(0, 1)\\) follow the directions below. (a) find the length of each segment. give an exact answer (not a decimal approximation). \\(ab = \\) \\(ef =\\)

Explanation:

Step1: Recall distance formula

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

Step2: Calculate length of \(AB\)

For \(A(-1,6)\) and \(B(4,8)\), \(x_1=-1,y_1 = 6,x_2 = 4,y_2=8\).
Substitute into formula: \(AB=\sqrt{(4 - (-1))^2+(8 - 6)^2}=\sqrt{(5)^2+(2)^2}=\sqrt{25 + 4}=\sqrt{29}\).

Step3: Calculate length of \(CD\)

For \(C(0,-3)\) and \(D(-2,-8)\), \(x_1 = 0,y_1=-3,x_2=-2,y_2=-8\).
Substitute: \(CD=\sqrt{(-2 - 0)^2+(-8 - (-3))^2}=\sqrt{(-2)^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}\).

Step4: Calculate length of \(EF\)

For \(E(-5,3)\) and \(F(0,1)\), \(x_1=-5,y_1 = 3,x_2 = 0,y_2=1\).
Substitute: \(EF=\sqrt{(0 - (-5))^2+(1 - 3)^2}=\sqrt{(5)^2+(-2)^2}=\sqrt{25 + 4}=\sqrt{29}\).

Answer:

\(AB=\sqrt{29}\), \(CD=\sqrt{29}\), \(EF=\sqrt{29}\)