Question

1. draw and label the image of figure mnop under the following sequence of transformations.

• dilation with center o and scale factor \\(\frac{2}{3}\\)
• reflection across line \\(\ell\\)
• translation along \\(\overrightarrow{jk}\\)

Explanation:

Step1: Perform Dilation (Center O, k=2/3)

For each vertex of MNOP:

• For point $O$: Since it's the center, $O' = O$
• For point $M$: Find the distance $OM$, multiply by $\frac{2}{3}$ to get $OM'$, so $M'$ lies on $\overrightarrow{OM}$ with $OM' = \frac{2}{3}OM$
• For point $N$: $ON' = \frac{2}{3}ON$, $N'$ on $\overrightarrow{ON}$
• For point $P$: $OP' = \frac{2}{3}OP$, $P'$ on $\overrightarrow{OP}$

Connect $M'N'O'P'$ for the dilated figure.

Step2: Reflect across line $\ell$

For each vertex of $M'N'O'P'$:

• Construct a perpendicular from the point to line $\ell$, measure the distance from the point to $\ell$.
• Mark the reflected point on the opposite side of $\ell$, at the same distance from $\ell$ as the original dilated point.

Label the reflected vertices $M''$, $N''$, $O''$, $P''$, connect them to form the reflected figure.

Step3: Translate along $\overrightarrow{JK}$

For each vertex of $M''N''O''P''$:

• Draw a vector equal to $\overrightarrow{JK}$ starting at each reflected vertex.
• The tip of each vector is the final transformed vertex: $M'''$, $N'''$, $O'''$, $P'''$.

Connect $M'''N'''O'''P'''$ for the final image.

Answer:

The final image is quadrilateral $M'''N'''O'''P'''$, created by first dilating MNOP by scale factor $\frac{2}{3}$ with center $O$, reflecting the result across line $\ell$, then translating the reflected figure along $\overrightarrow{JK}$. (To visualize, follow the step-by-step geometric transformations on the given diagram.)