Question

1. tyrell dilated a figure by a scale factor of \\(\frac{3}{2}\\). which of the following is a true statement?

a. the new figure will be smaller than the original figure. t
b. the perimeter of the new figure will be \\(\frac{3}{2}\\) times the perimeter of the original figure. t
c. the area of the new figure will be \\(\frac{3}{2}\\) times the area of the original figure. t
d. all of the above are true.

Explanation:

Step1: Analyze Scale Factor Impact on Size

The scale factor is $\frac{3}{2}$, which is greater than 1. So the new figure should be larger, not smaller. So option A is false.

Step2: Analyze Scale Factor Impact on Perimeter

For similar figures, the ratio of perimeters is equal to the scale factor. So perimeter of new figure = scale factor × perimeter of original = $\frac{3}{2}$ × original perimeter. So option B is true.

Step3: Analyze Scale Factor Impact on Area

For similar figures, the ratio of areas is the square of the scale factor. The square of $\frac{3}{2}$ is $\frac{9}{4}$, not $\frac{3}{2}$. So option C is false.

Step4: Evaluate Option D

Since A and C are false, D (all of the above) is also false. But wait, there's a mistake in Step3? Wait, no—wait, the problem's option C says "the area of the new figure will be $\frac{3}{2}$ times the area of the original figure"—but the correct ratio for area is scale factor squared, which is $(\frac{3}{2})^2=\frac{9}{4}$. But wait, maybe the problem has a typo? Wait, no, let's re - check. Wait, the scale factor is $\frac{3}{2}$. Perimeter ratio is $\frac{3}{2}$, area ratio is $(\frac{3}{2})^2=\frac{9}{4}$. But the options: B says perimeter is $\frac{3}{2}$ times original (correct, since perimeter scales with scale factor). A says new figure is smaller (wrong, $\frac{3}{2}>1$, so new is larger). C says area is $\frac{3}{2}$ times original (wrong, should be $\frac{9}{4}$). But the markings on the paper say B, C, D are marked as true? Wait, maybe I misread the scale factor. Wait, the scale factor is $\frac{3}{2}$? Wait, if the scale factor was less than 1, but $\frac{3}{2}$ is greater than 1. Wait, maybe the problem was supposed to be a scale factor less than 1? But as per the given problem, scale factor is $\frac{3}{2}$. But according to the markings on the paper, maybe there's a mistake in my analysis. Wait, no—wait, the user's problem: let's re - examine the options. Wait, the original problem: "Tyrell dilated a figure by a scale factor of $\frac{3}{2}$. Which of the following is a true statement? A. The new figure will be smaller... B. The perimeter... will be $\frac{3}{2}$ times... C. The area... will be $\frac{3}{2}$ times... D. All of the above." But according to the markings, B, C, D are marked. But mathematically, B is true (perimeter scales with scale factor), A is false (scale factor >1, so larger), C is false (area scales with square of scale factor, so $(\frac{3}{2})^2=\frac{9}{4}$). But maybe the problem had a scale factor of $\frac{1}{2}$? No, the problem says $\frac{3}{2}$. Wait, maybe the user made a typo, but assuming the problem as given:

Wait, no—wait, maybe I messed up. Wait, perimeter: scale factor is ratio of corresponding sides. So perimeter ratio is same as scale factor. So if scale factor is $\frac{3}{2}$, perimeter of new is $\frac{3}{2}$ times original. That's correct. Area: ratio is (scale factor)^2. So $(\frac{3}{2})^2=\frac{9}{4}$. But option C says $\frac{3}{2}$ times. So C is wrong. A is wrong (scale factor >1, so new is larger). So only B is true. But the markings on the paper have B, C, D marked. Maybe the scale factor was $\frac{1}{2}$? If scale factor was $\frac{1}{2}$ (less than 1):

- A: new figure smaller (true, since $\frac{1}{2}<1$)
- B: perimeter ratio $\frac{1}{2}$ (so perimeter of new is $\frac{1}{2}$ times original)
- C: area ratio $(\frac{1}{2})^2=\frac{1}{4}$ (but if option C was $\frac{1}{2}$ times, no—wait, no. Maybe the problem has a typo. But based on the given scale factor $\frac{3}{2}$:

But the question is to find the true statement. From the analysis, only B is true. But the markings on the paper might be incorrect, or maybe I misread the scale factor. Wait, the scale factor is written as $\frac{3}{2}$? Let me check the image again. The problem says "a scale factor of $\frac{3}{2}$". So with that, A is false, B is true, C is false, D is false. But the user's paper has B, C, D marked. Maybe there's a mistake in the problem or the markings. But based on the math:

Step1: Re - confirm Scale Factor

Scale factor $k = \frac{3}{2}>1$, so new figure is larger (A is false).

Step2: Perimeter Relationship

For similar figures, $\frac{\text{Perimeter of new}}{\text{Perimeter of original}}=k$. So $\text{Perimeter of new}=\frac{3}{2}\times\text{Perimeter of original}$ (B is true).

Step3: Area Relationship

For similar figures, $\frac{\text{Area of new}}{\text{Area of original}}=k^{2}=(\frac{3}{2})^{2}=\frac{9}{4}
eq\frac{3}{2}$ (C is false).

Step4: Evaluate D

Since A and C are false, D is false.

But maybe the scale factor was $\frac{1}{2}$? Let's check that case (maybe a typo). If $k = \frac{1}{2}<1$:

- A: new figure smaller (true)
- B: perimeter ratio $\frac{1}{2}$ (true, perimeter of new is $\frac{1}{2}$ times original)
- C: area ratio $(\frac{1}{2})^{2}=\frac{1}{4}$ (but if option C was $\frac{1}{2}$ times, no—wait, no. This is confusing. But based on the given problem with $k=\frac{3}{2}$:

Answer:

B. The perimeter of the new figure will be $\frac{3}{2}$ times the perimeter of the original figure.