using the given coordinates, determine whether $ \triangle abc \cong \triangle def $ by stating the length of each side.
if necessary round to the nearest tenth.
$ a(1,2), b(4,1), c(3,4), d(5,2), e(8,1), f(6,4) $
$ ab= $ $ de= $
$ bc= $ $ ef= $
$ ac= $ $ df= $
is $ \triangle abc \cong \triangle def $ ? (y/n)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of $AB$ and $DE$
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $AB$ with $A(1,2)$ and $B(4,1)$:
$$
\begin{align*}
AB&=\sqrt{(4 - 1)^2+(1 - 2)^2}\
&=\sqrt{3^2+(- 1)^2}\
&=\sqrt{9 + 1}\
&=\sqrt{10}\approx3.2
\end{align*}
$$
For $DE$ with $D(5,2)$ and $E(8,1)$:
$$
\begin{align*}
DE&=\sqrt{(8 - 5)^2+(1 - 2)^2}\
&=\sqrt{3^2+(-1)^2}\
&=\sqrt{9+1}\
&=\sqrt{10}\approx3.2
\end{align*}
$$
## Step2: Calculate the length of $BC$ and $EF$
For $BC$ with $B(4,1)$ and $C(3,4)$:
$$
\begin{align*}
BC&=\sqrt{(3 - 4)^2+(4 - 1)^2}\
&=\sqrt{(-1)^2+3^2}\
&=\sqrt{1 + 9}\
&=\sqrt{10}\approx3.2
\end{align*}
$$
For $EF$ with $E(8,1)$ and $F(6,4)$:
$$
\begin{align*}
EF&=\sqrt{(6 - 8)^2+(4 - 1)^2}\
&=\sqrt{(-2)^2+3^2}\
&=\sqrt{4 + 9}\
&=\sqrt{13}\approx3.6
\end{align*}
$$
## Step3: Calculate the length of $AC$ and $DF$
For $AC$ with $A(1,2)$ and $C(3,4)$:
$$
\begin{align*}
AC&=\sqrt{(3 - 1)^2+(4 - 2)^2}\
&=\sqrt{2^2+2^2}\
&=\sqrt{4 + 4}\
&=\sqrt{8}\approx2.8
\end{align*}
$$
For $DF$ with $D(5,2)$ and $F(6,4)$:
$$
\begin{align*}
DF&=\sqrt{(6 - 5)^2+(4 - 2)^2}\
&=\sqrt{1^2+2^2}\
&=\sqrt{1+4}\
&=\sqrt{5}\approx2.2
\end{align*}
$$

# Answer:
$AB = DE\approx3.2$, $BC\approx3.2$, $EF\approx3.6$, $AC\approx2.8$, $DF\approx2.2$, $N$ (Since not all corresponding sides are equal, $\triangle ABC$ is not congruent to $\triangle DEF$)

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of \(AB\) and \(DE\)

Step2: Calculate the length of \(BC\) and \(EF\)

Step3: Calculate the length of \(AC\) and \(DF\)

Answer: