Question

here are the endpoints of the segments $overline{ab}$, $overline{cd}$, and $overline{ef}$. a(7, 4), b(2, 8) c(5, 0), d(1, - 5) e(-3, 5), f(-8, 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 $A(7,4)$ and $B(2,8)$, we have $x_1 = 7,y_1 = 4,x_2=2,y_2 = 8$. Then $AB=\sqrt{(2 - 7)^2+(8 - 4)^2}=\sqrt{(-5)^2+4^2}=\sqrt{25 + 16}=\sqrt{41}$.

Step3: Calculate length of $CD$

For $C(5,0)$ and $D(1,-5)$, we have $x_1 = 5,y_1 = 0,x_2=1,y_2=-5$. Then $CD=\sqrt{(1 - 5)^2+(-5 - 0)^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}$.

Step4: Calculate length of $EF$

For $E(-3,5)$ and $F(-8,1)$, we have $x_1=-3,y_1 = 5,x_2=-8,y_2 = 1$. Then $EF=\sqrt{(-8+3)^2+(1 - 5)^2}=\sqrt{(-5)^2+(-4)^2}=\sqrt{25 + 16}=\sqrt{41}$.

Step5: Check congruence statements

Since $AB=\sqrt{41}$, $CD=\sqrt{41}$ and $EF=\sqrt{41}$, we have $\overline{AB}\cong\overline{CD}$, $\overline{AB}\cong\overline{EF}$ and $\overline{CD}\cong\overline{EF}$.

Answer:

(a) $AB=\sqrt{41}$
$CD=\sqrt{41}$
$EF=\sqrt{41}$
(b) $\overline{AB}\cong\overline{CD}$
$\overline{AB}\cong\overline{EF}$
$\overline{CD}\cong\overline{EF}$