the graph shows triangles mno and xyz. is mno congruent to xyz? justify your answer. yes, because a reflection across the x - axis followed by a rotation 90° counterclockwise around the origin maps mno onto xyz. yes, because a rotation 90° counterclockwise around m followed by a translation left 5 units and up 4 units maps mno onto xyz. no, because ( overline{mn} ) and ( overline{xy} ) do not have the same length. no, because ( overline{no} ) and ( overline{yz} ) do not have the same length.

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Accepted Answer

# Explanation:
## Step1: Calculate the length of $ \overline{MN} $
Use the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $. For points $ M(3,-2) $ and $ N(5,1) $, $ x_1 = 3,y_1=-2,x_2 = 5,y_2 = 1 $.
$$
\begin{align*}
d_{MN}&=\sqrt{(5 - 3)^2+(1+ 2)^2}\
&=\sqrt{2^2+3^2}\
&=\sqrt{4 + 9}\
&=\sqrt{13}
\end{align*}
$$
## Step2: Calculate the length of $ \overline{XY} $
For points $ X(-2,2) $ and $ Y(-5,4) $, $ x_1=-2,y_1 = 2,x_2=-5,y_2 = 4 $.
$$
\begin{align*}
d_{XY}&=\sqrt{(-5 + 2)^2+(4 - 2)^2}\
&=\sqrt{(-3)^2+2^2}\
&=\sqrt{9+4}\
&=\sqrt{13}
\end{align*}
$$
## Step3: Calculate the length of $ \overline{NO} $
For points $ N(5,1) $ and $ O(2,5) $, $ x_1 = 5,y_1 = 1,x_2=2,y_2 = 5 $.
$$
\begin{align*}
d_{NO}&=\sqrt{(2 - 5)^2+(5 - 1)^2}\
&=\sqrt{(-3)^2+4^2}\
&=\sqrt{9 + 16}\
&=\sqrt{25}\
&=5
\end{align*}
$$
## Step4: Calculate the length of $ \overline{YZ} $
For points $ Y(-5,4) $ and $ Z(-9,1) $, $ x_1=-5,y_1 = 4,x_2=-9,y_2 = 1 $.
$$
\begin{align*}
d_{YZ}&=\sqrt{(-9 + 5)^2+(1 - 4)^2}\
&=\sqrt{(-4)^2+(-3)^2}\
&=\sqrt{16+9}\
&=\sqrt{25}\
&=5
\end{align*}
$$
## Step5: Calculate the length of $ \overline{MO} $
For points $ M(3,-2) $ and $ O(2,5) $, $ x_1 = 3,y_1=-2,x_2=2,y_2 = 5 $.
$$
\begin{align*}
d_{MO}&=\sqrt{(2 - 3)^2+(5 + 2)^2}\
&=\sqrt{(-1)^2+7^2}\
&=\sqrt{1+49}\
&=\sqrt{50}=5\sqrt{2}
\end{align*}
$$
## Step6: Calculate the length of $ \overline{XZ} $
For points $ X(-2,2) $ and $ Z(-9,1) $, $ x_1=-2,y_1 = 2,x_2=-9,y_2 = 1 $.
$$
\begin{align*}
d_{XZ}&=\sqrt{(-9 + 2)^2+(1 - 2)^2}\
&=\sqrt{(-7)^2+(-1)^2}\
&=\sqrt{49+1}\
&=\sqrt{50}=5\sqrt{2}
\end{align*}
$$

Since $ MN = XY=\sqrt{13} $, $ NO=YZ = 5 $, $ MO=XZ=5\sqrt{2} $, by SSS (Side - Side - Side) congruence criterion, $ \triangle MNO\cong\triangle XYZ $.

# Answer:
Yes, because a reflection across the $ x $ - axis followed by a rotation $ 90^{\circ} $ counterclockwise around the origin maps $ MNO $ onto $ XYZ $.

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QUESTION IMAGE

Question

Explanation:

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

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

Step3: Calculate the length of \( \overline{NO} \)

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

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

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

Answer: