is def congruent to klm? justify your answer. yes, because a rotation 90° clockwise around d followed by a translation right 5 units maps def onto klm. yes, because a reflection across the y - axis followed by a rotation 180° around e maps def onto klm. no, because $\overline{de}$ and $\overline{kl}$ do not have the same length.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of $ \overline{DE} $
Use the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $. For $ D(-2,3) $ and $ E(-6,5) $, $ x_1=-2,y_1 = 3,x_2=-6,y_2 = 5 $.
$$
\begin{align*}
DE&=\sqrt{(-6+2)^2+(5 - 3)^2}\
&=\sqrt{(-4)^2+2^2}\
&=\sqrt{16 + 4}\
&=\sqrt{20}=2\sqrt{5}
\end{align*}
$$
## Step2: Calculate the length of $ \overline{KL} $
For $ K(3,2) $ and $ L(5,6) $, $ x_1=3,y_1 = 2,x_2=5,y_2 = 6 $.
$$
\begin{align*}
KL&=\sqrt{(5 - 3)^2+(6 - 2)^2}\
&=\sqrt{2^2+4^2}\
&=\sqrt{4+16}\
&=\sqrt{20}=2\sqrt{5}
\end{align*}
$$
## Step3: Calculate the length of $ \overline{EF} $
For $ E(-6,5) $ and $ F(-8,9) $, $ x_1=-6,y_1 = 5,x_2=-8,y_2 = 9 $.
$$
\begin{align*}
EF&=\sqrt{(-8 + 6)^2+(9 - 5)^2}\
&=\sqrt{(-2)^2+4^2}\
&=\sqrt{4+16}\
&=\sqrt{20}=2\sqrt{5}
\end{align*}
$$
## Step4: Calculate the length of $ \overline{LM} $
Assume $ M(10,7) $ (from the graph). For $ L(5,6) $ and $ M(10,7) $, $ x_1=5,y_1 = 6,x_2=10,y_2 = 7 $.
$$
\begin{align*}
LM&=\sqrt{(10 - 5)^2+(7 - 6)^2}\
&=\sqrt{5^2+1^2}\
&=\sqrt{25+1}\
&=\sqrt{26}
\end{align*}
$$
## Step5: Calculate the length of $ \overline{DF} $
For $ D(-2,3) $ and $ F(-8,9) $, $ x_1=-2,y_1 = 3,x_2=-8,y_2 = 9 $.
$$
\begin{align*}
DF&=\sqrt{(-8 + 2)^2+(9 - 3)^2}\
&=\sqrt{(-6)^2+6^2}\
&=\sqrt{36+36}\
&=\sqrt{72}=6\sqrt{2}
\end{align*}
$$
## Step6: Calculate the length of $ \overline{KM} $
For $ K(3,2) $ and $ M(10,7) $, $ x_1=3,y_1 = 2,x_2=10,y_2 = 7 $.
$$
\begin{align*}
KM&=\sqrt{(10 - 3)^2+(7 - 2)^2}\
&=\sqrt{7^2+5^2}\
&=\sqrt{49+25}\
&=\sqrt{74}
\end{align*}
$$

Since $ \overline{EF}
eq\overline{LM} $ and $ \overline{DF}
eq\overline{KM} $, the triangles are not congruent.

# Answer:
No, because $ \overline{DE} $ and $ \overline{KL} $ do not have the same length (also other corresponding sides are not equal as shown in the calculations).

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of \( \overline{DE} \)

Step2: Calculate the length of \( \overline{KL} \)

Step3: Calculate the length of \( \overline{EF} \)

Step4: Calculate the length of \( \overline{LM} \)

Step5: Calculate the length of \( \overline{DF} \)

Step6: Calculate the length of \( \overline{KM} \)

Answer: