Question

gabe and his classmates design a community greenhouse to grow vegetables throughout the year. the original plan for the floor is represented by figure ghij. gabe dilates the floor by a scale factor of 2/3 with a center of dilation at the origin to form figure ghij. then he translates figure ghij 3 units to the right and 3 units up. the final figure is labeled ghij. a. draw gabes transformations in the coordinate plane. b. what single transformation could map figure ghij onto figure ghij? c. why do you think the parallel sides in figure ghij remain parallel to the corresponding sides of figure ghij after the dilation?

Explanation:

Step1: Dilate the figure

For a dilation with scale - factor $k = \frac{2}{3}$ centered at the origin, if a point $(x,y)$ in figure $GHIJ$ has coordinates, the coordinates of the corresponding point $(x',y')$ in $G'H'I'J'$ are given by $(x',y')=(k x,k y)$.

Step2: Translate the dilated figure

For a translation of 3 units to the right and 3 units up, if a point $(x',y')$ in $G'H'I'J'$ has coordinates, the coordinates of the corresponding point $(x'',y'')$ in $G''H''I''J''$ are given by $(x'',y'')=(x'+3,y'+3)$.

Step3: Analyze single - transformation

To find a single transformation from $GHIJ$ to $G''H''I''J''$, we can consider the combined effect of dilation and translation. A single transformation would be a non - rigid transformation that first dilates and then translates.

Step4: Explain parallel sides

Dilation is a similarity transformation. Similarity transformations preserve angle measures. Since parallel lines have equal corresponding angles, and dilation preserves angle measures, parallel sides in figure $GHIJ$ remain parallel to the corresponding sides of figure $G'H'I'J'$.

Answer:

a. To draw the transformations: First, dilate each vertex of $GHIJ$ using the scale - factor $\frac{2}{3}$ centered at the origin. Then, take the dilated vertices and translate them 3 units to the right and 3 units up.
b. A single transformation that maps figure $GHIJ$ onto figure $G''H''I''J''$ is a non - rigid transformation that combines a dilation with a scale factor of $\frac{2}{3}$ centered at the origin and a translation of 3 units to the right and 3 units up.
c. Dilation is a similarity transformation that preserves angle measures. Parallel lines have equal corresponding angles, so parallel sides remain parallel after dilation.