Question

here are the endpoints of the segments $overline{ab}$, $overline{cd}$, and $overline{ef}$. $a(-8,1), b(-5,6)$ $c(-3,0), d(-7,-5)$ $e(5,3), f(7,-4)$ follow the directions below. (a) find the length of each segment. give an exact answer (not a decimal approximation). $ab = square$ $cd=square$ $ef=square$ (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 $A(-8,1)$ and $B(-5,6)$, $x_1=-8,y_1 = 1,x_2=-5,y_2 = 6$. Then $AB=\sqrt{(-5+8)^2+(6 - 1)^2}=\sqrt{3^2+5^2}=\sqrt{9 + 25}=\sqrt{34}$.

Step3: Calculate length of $CD$

For $C(-3,0)$ and $D(-7,-5)$, $x_1=-3,y_1 = 0,x_2=-7,y_2=-5$. Then $CD=\sqrt{(-7 + 3)^2+(-5-0)^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}$.

Step4: Calculate length of $EF$

For $E(5,3)$ and $F(7,-4)$, $x_1=5,y_1 = 3,x_2=7,y_2=-4$. Then $EF=\sqrt{(7 - 5)^2+(-4 - 3)^2}=\sqrt{2^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}$.

Step5: Check congruence statements

Since $\sqrt{34}
eq\sqrt{41}$, $\sqrt{34}
eq\sqrt{53}$, $\sqrt{41}
eq\sqrt{53}$, none of the segments are congruent.

Answer:

(a)
$AB=\sqrt{34}$
$CD=\sqrt{41}$
$EF=\sqrt{53}$
(b)
None of these are true