Question

practice 7 (from unit 5, lesson 5)
a circle with an area of 100π square centimeters is dilated so that its image has an area of 25π square centimeters. what is the scale factor of the dilation?
a 4
b 2
c \\(\frac{1}{2}\\)
d \\(\frac{1}{4}\\)

Explanation:

Step1: Recall the relationship between area and scale factor

For similar figures, if the scale factor of linear dimensions (radius, side length, etc.) is \( k \), the ratio of their areas is \( k^2 \). Let the original area be \( A_1 = 100\pi \) and the new area be \( A_2 = 25\pi \). The ratio of the areas is \( \frac{A_2}{A_1}=\frac{25\pi}{100\pi}=\frac{1}{4} \).

Step2: Find the scale factor \( k \)

Since the ratio of areas is \( k^2 \), we have \( k^2=\frac{1}{4} \). Taking the square root of both sides (and since dilation here is a reduction, we take the positive square root that is less than 1), we get \( k = \sqrt{\frac{1}{4}}=\frac{1}{2} \)? Wait, no, wait. Wait, \( \frac{A_2}{A_1}=\frac{25\pi}{100\pi}=\frac{1}{4} \), so \( k^2=\frac{1}{4} \), so \( k = \frac{1}{2} \)? Wait, no, wait, the scale factor for area is the square of the linear scale factor. Wait, original area is \( 100\pi \), new area is \( 25\pi \). So the ratio of areas is \( \frac{25\pi}{100\pi}=\frac{1}{4} \). So if the linear scale factor is \( k \), then \( k^2=\frac{1}{4} \), so \( k=\frac{1}{2} \)? Wait, no, wait, maybe I mixed up. Wait, the original circle has area \( 100\pi \), so its radius \( r_1 \) satisfies \( \pi r_1^2 = 100\pi \), so \( r_1^2 = 100 \), \( r_1 = 10 \). The new circle has area \( 25\pi \), so its radius \( r_2 \) satisfies \( \pi r_2^2 = 25\pi \), so \( r_2^2 = 25 \), \( r_2 = 5 \). Then the scale factor \( k=\frac{r_2}{r_1}=\frac{5}{10}=\frac{1}{2} \)? Wait, no, wait, \( 5/10 = 1/2 \), but wait, the area ratio is \( (1/2)^2 = 1/4 \), which matches \( 25\pi/100\pi = 1/4 \). Wait, but the options have \( \frac{1}{2} \) and \( \frac{1}{4} \). Wait, no, wait, maybe I made a mistake. Wait, no, let's recalculate. Wait, original area \( A_1 = 100\pi \), new area \( A_2 = 25\pi \). The ratio of areas is \( A_2/A_1 = 25\pi/100\pi = 1/4 \). The scale factor for area is the square of the linear scale factor. So if the linear scale factor is \( k \), then \( k^2 = A_2/A_1 = 1/4 \), so \( k = \sqrt{1/4} = 1/2 \)? Wait, but \( (1/2)^2 = 1/4 \), which is the area ratio. But wait, the original radius is 10, new radius is 5, so the scale factor from original to new is 5/10 = 1/2. So the scale factor of dilation is 1/2? But wait, the options: C is 1/2, D is 1/4. Wait, maybe I messed up. Wait, let's check again. Original area: \( \pi r_1^2 = 100\pi \implies r_1 = 10 \). New area: \( \pi r_2^2 = 25\pi \implies r_2 = 5 \). So the scale factor \( k = r_2 / r_1 = 5/10 = 1/2 \). So the scale factor is 1/2, which is option C? Wait, but wait, the area ratio is (1/2)^2 = 1/4, which matches 25π/100π = 1/4. So yes, the scale factor is 1/2. Wait, but let's check the options again. Option C is 1/2, option D is 1/4. So the correct answer is C? Wait, no, wait, maybe I had it backwards. Wait, dilation: if the image is smaller, the scale factor is less than 1. Original area is 100π, image area is 25π, so it's a reduction. The scale factor is the ratio of the new linear dimension to the original. So radius goes from 10 to 5, so scale factor is 5/10 = 1/2. So the answer is C.

Answer:

C. \(\frac{1}{2}\)