Question

question
the rectangle on the right is a scaled copy of the rectangle on the left. identify the scale factor.
express your answer as a whole number or fraction in simplest form.
30
12
60
24

Explanation:

Step1: Recall scale factor formula

The scale factor is the ratio of a length in the scaled copy to the corresponding length in the original figure. So, we can take the height (or width) of the right rectangle and divide it by the height (or width) of the left rectangle. Let's use the heights first. The height of the left rectangle is \( 12 \), and the height of the right rectangle is \( 60 \).

Step2: Calculate the scale factor

Using the formula for scale factor \( k=\frac{\text{length in scaled copy}}{\text{length in original}} \), we substitute the values: \( k = \frac{60}{12} \). Simplifying this fraction, \( \frac{60\div12}{12\div12}=\frac{5}{1} = 5 \)? Wait, no, wait. Wait, maybe I mixed up original and scaled. Wait, the right is the scaled copy of the left. Wait, maybe I should check the widths too. The width of the left rectangle is \( 30 \), the width of the right is \( 24 \)? Wait, no, that can't be. Wait, maybe I misread the labels. Wait, the left rectangle has height \( 12 \) and width \( 30 \). The right rectangle has height \( 60 \) and width \( 24 \)? Wait, no, that would be inconsistent. Wait, maybe the left rectangle has width \( 30 \) and height \( 12 \), and the right has width \( 24 \) and height \( 60 \)? Wait, no, that would mean different scale factors for width and height, which can't be. Wait, maybe I made a mistake. Wait, no, wait, maybe the left rectangle's width is \( 30 \), and the right's width is \( 24 \)? No, that would be a reduction, but height is an increase. Wait, no, maybe the left is the original, right is scaled. Wait, let's check both dimensions. Let's take height: left height is \( 12 \), right height is \( 60 \). So \( 60\div12 = 5 \). Width: left width is \( 30 \), right width is \( 24 \). \( 24\div30=\frac{4}{5} \). Wait, that's a problem. Wait, maybe I mislabeled the rectangles. Wait, maybe the left rectangle has height \( 12 \) and width \( 30 \), and the right has height \( 60 \) and width \( 150 \)? No, the diagram shows right rectangle with width \( 24 \). Wait, maybe the left rectangle's height is \( 12 \), right is \( 60 \) (so scale factor \( 60/12 = 5 \)), but width: left is \( 30 \), right should be \( 30\times5 = 150 \), but the diagram says right width is \( 24 \). Wait, that's a contradiction. Wait, maybe I mixed up the rectangles. Wait, maybe the left rectangle has height \( 12 \) and width \( 30 \), and the right has height \( 60 \) and width \( 24 \)? No, that can't be. Wait, maybe the left rectangle's height is \( 12 \), right is \( 60 \) (scale factor 5), but width: left is \( 30 \), right is \( 24 \), which is \( 24/30 = 4/5 \). That's inconsistent. Wait, maybe the labels are wrong. Wait, maybe the left rectangle has width \( 30 \) and height \( 12 \), and the right has width \( 24 \) and height \( 60 \)? No, the right's width is \( 24 \), height \( 60 \). So height scale factor: \( 60/12 = 5 \), width scale factor: \( 24/30 = 4/5 \). That's impossible. Wait, maybe I made a mistake in the problem. Wait, no, maybe the left rectangle is the original, and the right is scaled, but maybe the width of the left is \( 30 \), and the width of the right is \( 24 \), so scale factor for width is \( 24/30 = 4/5 \), and height: left is \( 12 \), right is \( 60 \), scale factor \( 60/12 = 5 \). That's a problem. Wait, maybe the rectangles are swapped? Wait, maybe the left rectangle has height \( 60 \) and width \( 24 \), and the right has height \( 12 \) and width \( 30 \)? No, the problem says "the rectangle on the right is a scaled copy of the left". So right is scaled from left. So all corresponding sides must have the same scale factor. So maybe I misread the numbers. Wait, looking at the diagram again: left rectangle: height \( 12 \), width \( 30 \). Right rectangle: height \( 60 \), width \( 24 \). Wait, that's impossible. Wait, maybe the right rectangle's width is \( 150 \), but the diagram says \( 24 \). Wait, no, the user's diagram shows right rectangle with width \( 24 \) and height \( 60 \), left with width \( 30 \) and height \( 12 \). Wait, maybe the problem has a typo, but assuming that we take the height: left height \( 12 \), right height \( 60 \), so scale factor \( 60/12 = 5 \), but width: left \( 30 \), right should be \( 30\times5 = 150 \), but it's \( 24 \). Alternatively, maybe the width: left \( 30 \), right \( 24 \), scale factor \( 24/30 = 4/5 \), and height: left \( 12 \), right \( 60 \), scale factor \( 60/12 = 5 \). That's a contradiction. Wait, maybe I mixed up height and width. Wait, maybe the left rectangle's width is \( 12 \) and height \( 30 \), and the right's width is \( 24 \) and height \( 60 \). Then width scale factor: \( 24/12 = 2 \), height scale factor: \( 60/30 = 2 \). Ah! That makes sense. So maybe I misread the labels. So left rectangle: width \( 12 \), height \( 30 \). Right rectangle: width \( 24 \), height \( 60 \). Then scale factor is \( 24/12 = 2 \) (or \( 60/30 = 2 \)). Wait, the original problem's diagram: left rectangle has "30" on top (width) and "12" on the side (height). Right rectangle has "24" on the bottom (width) and "60" on the side (height). Wait, that's the issue. So left width: 30, left height: 12. Right width: 24, right height: 60. So width scale factor: 24/30 = 4/5, height scale factor: 60/12 = 5. That's inconsistent. So there must be a mistake in my reading. Wait, maybe the left rectangle's height is 12, right is 60 (scale factor 5), and left width is 30, right width is 30*5=150, but the diagram says 24. So maybe the diagram is labeled incorrectly, or I misread. Wait, maybe the left rectangle has width 30 and height 12, and the right has width 24 and height 60, but that's a mistake. Alternatively, maybe the right rectangle is a scaled copy, but we take the width: 24/30 = 4/5, and height: 60/12 = 5. That's impossible. Wait, maybe the problem is that the left rectangle is the scaled copy of the right? But the problem says "the rectangle on the right is a scaled copy of the left". So right is scaled from left. So all sides must scale by the same factor. So maybe the numbers are different. Wait, maybe the left rectangle has height 12, right has height 60 (scale factor 5), and left width 30, right width 30*5=150, but the diagram shows 24. So maybe the diagram has a typo, and the right width is 150. But the user's diagram shows 24. Wait, maybe I made a mistake. Wait, let's check the problem again. The user's diagram: left rectangle with top labeled 30, side labeled 12. Right rectangle with bottom labeled 24, side labeled 60. So left: width 30, height 12. Right: width 24, height 60. So scale factor for height: 60/12 = 5. Scale factor for width: 24/30 = 4/5. That's a problem. So maybe the labels are swapped. Maybe left rectangle has width 12 and height 30, right has width 24 and height 60. Then scale factor: 24/12 = 2, 60/30 = 2. That works. So maybe the "30" is the height, and "12" is the width. So left width: 12, left height: 30. Right width: 24, right height: 60. Then scale factor is 24/12 = 2 (or 60/30 = 2). So that must be the case. So the problem was mislabeled in my initial reading. So we take the width: right width (24) divided by left width (12) = 2. Or height: right height (60) divided by left height (30) = 2. So scale factor is 2. Wait, but 60/12 is 5, but 30*2=60? No, 30*2=60, 12*2=24. Ah! There we go. So left rectangle: width 12, height 30. Right rectangle: width 24, height 60. So 12*2=24, 30*2=60. So scale factor is 2. So my initial mistake was reading the left rectangle's width as 30, but it's actually the height. So the left rectangle has width 12 (side) and height 30 (top). So scale factor is 24/12 = 2 (or 60/30 = 2).

Step1: Identify corresponding sides

Let's correctly identify the corresponding sides. Let the left rectangle have width \( w_1 = 12 \) and height \( h_1 = 30 \). The right rectangle (scaled copy) has width \( w_2 = 24 \) and height \( h_2 = 60 \).

Step2: Calculate scale factor

The scale factor \( k \) is the ratio of a side in the scaled copy to the corresponding side in the original. Using the width: \( k=\frac{w_2}{w_1}=\frac{24}{12} \). Simplifying, \( \frac{24}{12}=2 \). We can verify with the height: \( \frac{h_2}{h_1}=\frac{60}{30}=2 \). Both give the same scale factor, so it's consistent.

Answer:

\( 2 \)