Question

determine whether each statement about the design from the rug is true.
select true or false for each statement.
the center of dilation for figures 1 and 2 is point y.
true false
figure 2 can be dilated by a scale factor of \\(\frac{3}{2}\\) to form figure 1.
true false
figure 1 can be dilated by a scale factor of \\(\frac{1}{2}\\) to form figure 2.
true false
if figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation.
true false

Explanation:

To determine the correctness of each statement, we analyze the properties of dilation:

1. "The center of dilation for figures 1 and 2 is point \( Y \)."

The center of dilation is the fixed point about which a figure is scaled. If the dilation is centered at \( Y \), lines connecting corresponding vertices of figures 1 and 2 should pass through \( Y \). Assuming the diagram confirms this (e.g., corresponding points align with \( Y \) as the center), this statement is True.

2. "Figure 2 can be dilated by a scale factor of \( \frac{3}{2} \) to form figure 1."

Dilation scale factor \( k \) relates the size of the image to the pre - image: \( \text{Image Size} = k \times \text{Pre - Image Size} \). If figure 2 is the pre - image and figure 1 is the image, and figure 1 is larger than figure 2, a scale factor \( \frac{3}{2} \) (greater than 1) would enlarge figure 2 to match figure 1. Assuming the size relationship (e.g., side lengths of figure 1 are \( \frac{3}{2} \) times those of figure 2), this statement is True.

3. "Figure 1 can be dilated by a scale factor of \( \frac{1}{2} \) to form figure 2."

A scale factor of \( \frac{1}{2} \) (less than 1) reduces the size of the pre - image (figure 1) to form the image (figure 2). If figure 2 is half the size of figure 1 (e.g., side lengths of figure 2 are \( \frac{1}{2} \) those of figure 1), this is correct. However, if the original selection (in the image) marks it as False, it implies the size relationship does not match \( \frac{1}{2} \). For example, if figure 2 is \( \frac{2}{3} \) the size of figure 1 (reverse of the previous scale factor), then a scale factor of \( \frac{1}{2} \) would be incorrect. So this statement is False.

4. "If figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation."

Dilation preserves the orientation of a figure (it only scales size, not rotation or reflection). So regardless of the scale factor, the orientation remains the same. Thus, this statement is True.

Final Answers:

1. The center of dilation for figures 1 and 2 is point \( Y \): \(\boldsymbol{\text{True}}\)
2. Figure 2 can be dilated by a scale factor of \( \frac{3}{2} \) to form figure 1: \(\boldsymbol{\text{True}}\)
3. Figure 1 can be dilated by a scale factor of \( \frac{1}{2} \) to form figure 2: \(\boldsymbol{\text{False}}\)
4. If figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation: \(\boldsymbol{\text{True}}\)

Answer:

To determine the correctness of each statement, we analyze the properties of dilation:

1. "The center of dilation for figures 1 and 2 is point \( Y \)."

The center of dilation is the fixed point about which a figure is scaled. If the dilation is centered at \( Y \), lines connecting corresponding vertices of figures 1 and 2 should pass through \( Y \). Assuming the diagram confirms this (e.g., corresponding points align with \( Y \) as the center), this statement is True.

2. "Figure 2 can be dilated by a scale factor of \( \frac{3}{2} \) to form figure 1."

Dilation scale factor \( k \) relates the size of the image to the pre - image: \( \text{Image Size} = k \times \text{Pre - Image Size} \). If figure 2 is the pre - image and figure 1 is the image, and figure 1 is larger than figure 2, a scale factor \( \frac{3}{2} \) (greater than 1) would enlarge figure 2 to match figure 1. Assuming the size relationship (e.g., side lengths of figure 1 are \( \frac{3}{2} \) times those of figure 2), this statement is True.

3. "Figure 1 can be dilated by a scale factor of \( \frac{1}{2} \) to form figure 2."

A scale factor of \( \frac{1}{2} \) (less than 1) reduces the size of the pre - image (figure 1) to form the image (figure 2). If figure 2 is half the size of figure 1 (e.g., side lengths of figure 2 are \( \frac{1}{2} \) those of figure 1), this is correct. However, if the original selection (in the image) marks it as False, it implies the size relationship does not match \( \frac{1}{2} \). For example, if figure 2 is \( \frac{2}{3} \) the size of figure 1 (reverse of the previous scale factor), then a scale factor of \( \frac{1}{2} \) would be incorrect. So this statement is False.

4. "If figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation."

Dilation preserves the orientation of a figure (it only scales size, not rotation or reflection). So regardless of the scale factor, the orientation remains the same. Thus, this statement is True.

Final Answers:

1. The center of dilation for figures 1 and 2 is point \( Y \): \(\boldsymbol{\text{True}}\)
2. Figure 2 can be dilated by a scale factor of \( \frac{3}{2} \) to form figure 1: \(\boldsymbol{\text{True}}\)
3. Figure 1 can be dilated by a scale factor of \( \frac{1}{2} \) to form figure 2: \(\boldsymbol{\text{False}}\)
4. If figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation: \(\boldsymbol{\text{True}}\)