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

Question

Accepted Answer

# Explanation:
## Step1: Recall the distance formula
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
## Step2: Calculate the length of $AB$
For $A(3,1)$ and $B(-2,-5)$, $x_1 = 3,y_1 = 1,x_2=-2,y_2=-5$.
$$
\begin{align*}
AB&=\sqrt{(-2 - 3)^2+(-5 - 1)^2}\
&=\sqrt{(-5)^2+(-6)^2}\
&=\sqrt{25 + 36}\
&=\sqrt{61}
\end{align*}
$$
## Step3: Calculate the length of $CD$
For $C(-8,4)$ and $D(-4,-3)$, $x_1=-8,y_1 = 4,x_2=-4,y_2=-3$.
$$
\begin{align*}
CD&=\sqrt{(-4+8)^2+(-3 - 4)^2}\
&=\sqrt{(4)^2+(-7)^2}\
&=\sqrt{16 + 49}\
&=\sqrt{65}
\end{align*}
$$
## Step4: Calculate the length of $EF$
For $E(-7,8)$ and $F(-2,2)$, $x_1=-7,y_1 = 8,x_2=-2,y_2=2$.
$$
\begin{align*}
EF&=\sqrt{(-2 + 7)^2+(2 - 8)^2}\
&=\sqrt{(5)^2+(-6)^2}\
&=\sqrt{25+36}\
&=\sqrt{61}
\end{align*}
$$
## Step5: Check the congruent segments
Since $AB=\sqrt{61}$ and $EF=\sqrt{61}$, so $\overline{AB}\cong\overline{EF}$

# Answer:
(a) $AB=\sqrt{61}$, $CD=\sqrt{65}$, $EF=\sqrt{61}$
(b) $\overline{AB}\cong\overline{EF}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall the distance formula

Step2: Calculate the length of \(AB\)

Step3: Calculate the length of \(CD\)

Step4: Calculate the length of \(EF\)

Step5: Check the congruent segments

Answer: