Question

What transformations figures $1$ and $2$ require to form figure $3$?

• A. Figure 1: Rotation ${180}^o$ counterclockwise about the origin, translation;
  Figure 2: Dilation about the origin with a scale factor of $0.5$, reflection on the y-axis, translation
• B. Figure 1: Rotation ${90}^o$ clockwise about the origin, translation;
  Figure 2: Dilation about the origin with a scale factor of $0.5$, reflection on the y-axis, translation
• C. Figure 1: Rotation ${90}^o$ counterclockwise about the origin, translation;
  Figure 2: Dilation about the origin with a scale factor of $2$, reflection on the y-axis, translation
• D. Figure 1: Rotation ${90}^o$ counterclockwise about the origin, translation;
  Figure 2: Dilation about the origin with a scale factor of $0.5$, reflection on the y-axis, translation

Answer 1

The correct option is D Figure 1: Rotation ${90}^o$ counterclockwise about the origin, translation;

Figure 2: Dilation about the origin with a scale factor of $0.5$, reflection on the y-axis, translation
Let’s observe both the preimage and the image and go through the options one-by-one.

Here, in option A, we notice that figure 2 undergoes dilation about the origin with a scale factor of $2$. This means rectangle $E^{\prime }{F}^{\prime }{G}^{\prime }{H}^{\prime }$ is twice the size of rectangle $EFGH,$ which is not the desired image. Hence, option A is the incorrect option.

In option B, figure 1 undergoes rotation of 90° counterclockwise about the origin. This means the image $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ lies in the second quadrant, which is the desired image. Similarly, figure 2 undergoes dilation about the origin with a scale factor of $0.5$. This means rectangle $E^{\prime }{F}^{\prime }{G}^{\prime }{H}^{\prime }$ is half the size of rectangle $EFGH$, which is also the desired image. Hence, option B may be the correct option.

In option C, figure 1 undergoes rotation of ${180}^o$ counterclockwise about the origin. This means the rectangle $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ lies in the third quadrant, which is not the desired image. Hence, option C is the incorrect option.

In option D, figure 1 undergoes rotation of ${90}^o$ clockwise about the origin. This means the rectangle $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ lies in the fourth quadrant, which is not the desired image. Hence, option D is the incorrect option.

Step 1:
Figure 1: Rotation ${90}^o$ counterclockwise about the origin

Step 2:
Figure 1: Translation of $1$ unit toward the left and then $2$ units top

Step 3:
Figure 2: Dilation about the origin with a scale factor of $0.5$

Step 4:
Figure 2: Reflection on the $y$-axis
Step 5: Translation of $2$ units toward the left and then of $1$ unit up
Hence, the correct answer is option B.