Question

segments and angles identifying congruent segments in the plane here are the endpoints of the segments $overline{ab}$, $overline{cd}$, and $overline{ef}$. a(1, - 7), b(-3, - 8) c(3, 2), d(2, - 1) e(-5, 4), f(-7, 1) follow the directions below. (a) find the length of each segment. give an exact answer (not a decimal approximation). ab = cd = ef = (b) check all statements that are true below. $squareoverline{ab}congoverline{cd}$ $squareoverline{ab}congoverline{ef}$ $squareoverline{cd}congoverline{ef}$ $square$ none of these are true

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 $A(1,-7)$ and $B(-3,-8)$, we have $x_1 = 1,y_1=-7,x_2=-3,y_2 = - 8$. Then $AB=\sqrt{(-3 - 1)^2+(-8+7)^2}=\sqrt{(-4)^2+(-1)^2}=\sqrt{16 + 1}=\sqrt{17}$.

Step3: Calculate length of $CD$

For $C(3,2)$ and $D(2,-1)$, we have $x_1 = 3,y_1 = 2,x_2=2,y_2=-1$. Then $CD=\sqrt{(2 - 3)^2+(-1 - 2)^2}=\sqrt{(-1)^2+(-3)^2}=\sqrt{1 + 9}=\sqrt{10}$.

Step4: Calculate length of $EF$

For $E(-5,4)$ and $F(-7,1)$, we have $x_1=-5,y_1 = 4,x_2=-7,y_2=1$. Then $EF=\sqrt{(-7 + 5)^2+(1 - 4)^2}=\sqrt{(-2)^2+(-3)^2}=\sqrt{4+9}=\sqrt{13}$.

Step5: Check congruence

Since $\sqrt{17}
eq\sqrt{10}
eq\sqrt{13}$, none of the segments are congruent.

Answer:

(a) $AB=\sqrt{17}$, $CD=\sqrt{10}$, $EF=\sqrt{13}$
(b) None of these are true