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) $
$ \frac{ab}{de}=\frac{3.2}{} $
$ \frac{bc}{ef}=\frac{}{} $
$ \frac{ac}{df}=\frac{}{} $
is $ \triangle abc \cong \triangle def $ ? (y/n) $ \frac{}{} $

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of $AC$
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $A(1,2)$ and $C(3,4)$, $x_1 = 1,y_1 = 2,x_2=3,y_2 = 4$.
$$
\begin{align*}
AC&=\sqrt{(3 - 1)^2+(4 - 2)^2}\
&=\sqrt{2^2+2^2}\
&=\sqrt{4 + 4}\
&=\sqrt{8}\approx2.8
\end{align*}
$$
## Step2: Calculate the length of $BC$
For $B(4,1)$ and $C(3,4)$, $x_1 = 4,y_1 = 1,x_2=3,y_2 = 4$.
$$
\begin{align*}
BC&=\sqrt{(3 - 4)^2+(4 - 1)^2}\
&=\sqrt{(-1)^2+3^2}\
&=\sqrt{1 + 9}\
&=\sqrt{10}\approx3.2
\end{align*}
$$
## Step3: Calculate the length of $DE$
For $D(5,2)$ and $E(8,1)$, $x_1 = 5,y_1 = 2,x_2=8,y_2 = 1$.
$$
\begin{align*}
DE&=\sqrt{(8 - 5)^2+(1 - 2)^2}\
&=\sqrt{3^2+(-1)^2}\
&=\sqrt{9+1}\
&=\sqrt{10}\approx3.2
\end{align*}
$$
## Step4: Calculate the length of $EF$
For $E(8,1)$ and $F(6,4)$, $x_1 = 8,y_1 = 1,x_2=6,y_2 = 4$.
$$
\begin{align*}
EF&=\sqrt{(6 - 8)^2+(4 - 1)^2}\
&=\sqrt{(-2)^2+3^2}\
&=\sqrt{4 + 9}\
&=\sqrt{13}\approx3.6
\end{align*}
$$
## Step5: Calculate the length of $DF$
For $D(5,2)$ and $F(6,4)$, $x_1 = 5,y_1 = 2,x_2=6,y_2 = 4$.
$$
\begin{align*}
DF&=\sqrt{(6 - 5)^2+(4 - 2)^2}\
&=\sqrt{1^2+2^2}\
&=\sqrt{1+4}\
&=\sqrt{5}\approx2.2
\end{align*}
$$
## Step6: Check for congruence
Since $AB = 3.2$, $DE=3.2$, $BC = 3.2$, $EF = 3.6$, $AC=2.8$, $DF = 2.2$. The sides do not satisfy the SSS (Side - Side - Side) congruence criterion.

# Answer:
$AC = 2.8$, $BC=3.2$, $DE = 3.2$, $EF = 3.6$, $DF=2.2$, $\triangle ABC
ot\cong\triangle DEF$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of \(AC\)

Step2: Calculate the length of \(BC\)

Step3: Calculate the length of \(DE\)

Step4: Calculate the length of \(EF\)

Step5: Calculate the length of \(DF\)

Step6: Check for congruence

Answer: