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 $\frac{2}{3}$ 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?

Explanation:

Step1: Perform dilation

To dilate a point $(x,y)$ by a scale factor $k = \frac{2}{3}$ with center of dilation at the origin $(0,0)$, we use the rule $(x,y)\to(kx,ky)$. For example, if a vertex of $GHIJ$ has coordinates $(x_1,y_1)$, the coordinates of the corresponding vertex of $G'H'I'J'$ will be $(\frac{2}{3}x_1,\frac{2}{3}y_1)$.

Step2: Perform translation

To translate a point $(x,y)$ 3 units to the right and 3 units up, we use the rule $(x,y)\to(x + 3,y+ 3)$. So if a vertex of $G'H'I'J'$ has coordinates $(x_2,y_2)$, the coordinates of the corresponding vertex of $G''H''I''J''$ will be $(x_2 + 3,y_2+3)$.

Step3: Analyze single - transformation

Let the original point be $(x,y)$. After dilation by a scale factor $\frac{2}{3}$ with center at the origin, the point becomes $(\frac{2}{3}x,\frac{2}{3}y)$. Then after translation 3 units right and 3 units up, it becomes $(\frac{2}{3}x+3,\frac{2}{3}y + 3)$. We can think of this as a non - standard transformation. However, if we consider the general form of a transformation. A dilation followed by a translation is a similarity transformation. We can rewrite the transformation from $(x,y)$ to $(\frac{2}{3}x+3,\frac{2}{3}y + 3)$ as a composition of a dilation and a translation. But if we want a single transformation, we can consider it as a transformation of the form $(x,y)\to(\frac{2}{3}x+3,\frac{2}{3}y + 3)$ which is a type of affine transformation. In terms of common geometric transformations, we can say it is a non - rigid transformation that combines dilation and translation.

Answer:

a. To draw the transformations:

• First, find the coordinates of the vertices of $G'H'I'J'$ by dilating the vertices of $GHIJ$ using the scale factor $\frac{2}{3}$ with center at the origin.
• Then, find the coordinates of the vertices of $G''H''I''J''$ by translating the vertices of $G'H'I'J'$ 3 units to the right and 3 units up. Plot all the figures on the coordinate plane.

b. A non - rigid transformation (a combination of dilation and translation) that maps $(x,y)$ to $(\frac{2}{3}x+3,\frac{2}{3}y + 3)$ can map figure $GHIJ$ onto figure $G''H''I''J''$.