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.

Explanation:

Step1: Recall scale factor formula

The scale factor from the original (left rectangle) to the scaled copy (right rectangle) is the ratio of corresponding side lengths. Let's take the vertical sides: original length is 6, scaled length is 21. Or horizontal sides: original is 14, scaled is 9? Wait, no, wait. Wait, maybe I mixed up. Wait, the left rectangle has height 6 and width 14. The right rectangle has height 21 and width 9? Wait, no, that can't be. Wait, maybe the right rectangle's width corresponds to left's height? Wait, no, scaled copy should have proportional sides. Wait, maybe I made a mistake. Wait, no, the scale factor is (scaled length) / (original length). Let's check vertical sides: left height is 6, right height is 21. So 21/6 = 7/2? Wait, but horizontal: left width 14, right width 9? That doesn't match. Wait, maybe I got the correspondence wrong. Wait, maybe the left rectangle's width is 14 and height 6, and the right rectangle's width is 9 and height 21? That would mean that the scale factor for height is 21/6 = 7/2, and for width is 9/14, which is not the same. That can't be. Wait, maybe the labels are wrong. Wait, maybe the left rectangle has width 14 and height 6, and the right rectangle has width (let's see, if scale factor is k, then 14*k should be the width of right, and 6*k should be the height of right. Wait, the right rectangle's height is 21, so 6*k = 21 => k = 21/6 = 7/2. Then the width of right should be 14*(7/2) = 49, but the right rectangle's width is 9? That's not possible. Wait, maybe I mixed up the rectangles. Wait, maybe the left is the scaled copy and the right is original? No, the problem says "the rectangle on the right is a scaled copy of the left". So right is scaled from left. So corresponding sides: left height 6, right height 21. So scale factor is 21/6 = 7/2? But then left width 14, right width should be 14*(7/2)=49, but the right has width 9. That's a problem. Wait, maybe the horizontal sides: left width 14, right width 9. Then scale factor is 9/14, and height: left 6, right 21: 21/6=7/2. That's inconsistent. So maybe I misread the diagram. Wait, maybe the left rectangle has height 6 and width 14, and the right rectangle has height 21 and width (14*(21/6))=49? But the diagram shows right width as 9. Wait, maybe the diagram is labeled with right width 9 and height 21, and left width 14 and height 6. That would mean that the scale factor is (9/14) for width and (21/6)=7/2 for height, which is impossible. So maybe there's a mistake in my reading. Wait, maybe the left rectangle's height is 6, right's height is 21: 21/6 = 3.5 = 7/2. Left width 14, right width: 14*(7/2)=49. But the diagram shows right width as 9. So maybe the diagram is flipped? Maybe the left rectangle's width is 6 and height 14? No, the left has 6 on the side (vertical) and 14 on the top (horizontal). So vertical side is 6 (height), horizontal is 14 (width). Right rectangle has vertical side 21 (height) and horizontal side 9 (width). That can't be a scaled copy. Wait, maybe the right rectangle's width is 49 and height 21, but the diagram has 9. Maybe the diagram is wrong, or I misread. Wait, maybe the problem is that the left rectangle has height 6 and width 14, and the right rectangle has height 21 and width (14*(21/6))=49, but the diagram shows 9. That's a problem. Wait, maybe I made a mistake. Wait, let's check again. The problem says "the rectangle on the right is a scaled copy of the rectangle on the left". So corresponding sides must be proportional. So let's take the vertical sides: left height 6, right height 21. So scale factor k = 21/6 = 7/2. Then the horizontal side of the right rectangle should be 14*k = 14*(7/2) = 49. But the diagram shows 9. That's inconsistent. Alternatively, maybe the horizontal sides: left width 14, right width 9. Then k = 9/14, and vertical side of right should be 6*(9/14) = 27/7, which is not 21. So that's impossible. Wait, maybe the labels are swapped. Maybe the left rectangle has height 14 and width 6, and the right has height 21 and width 9. Then vertical scale factor: 21/14 = 3/2, horizontal scale factor: 9/6 = 3/2. Ah! That makes sense. Oh! I see. I misread the left rectangle's height and width. The left rectangle: the vertical side is 6? No, wait, the left rectangle has 6 on the vertical (height) and 14 on horizontal (width). But if the left rectangle's height is 14 and width is 6, then right height 21 (21/14=3/2) and right width 9 (9/6=3/2). That works. So maybe the diagram's labels are swapped. So the left rectangle: height 14, width 6. Right rectangle: height 21, width 9. Then scale factor is 21/14 = 3/2, and 9/6 = 3/2. So that's consistent. So I must have misread the left rectangle's dimensions. The left rectangle: vertical side (height) is 6? No, the number 6 is on the left side (vertical), so height is 6, width is 14. Then right height 21, right width... Wait, maybe the right rectangle's width is (14)*(21/6) = 49, but the diagram shows 9. That's a problem. Wait, maybe the problem is that the left rectangle has width 6 and height 14, and the right has width 9 and height 21. Then scale factor for width: 9/6 = 3/2, scale factor for height: 21/14 = 3/2. Ah! That's it. I misread the left rectangle's width and height. The left rectangle: width 6 (horizontal) and height 14 (vertical). Then right rectangle: width 9 (horizontal) and height 21 (vertical). Then scale factor is 9/6 = 3/2 and 21/14 = 3/2. Yes, that makes sense. So the left rectangle has width 6 and height 14, right has width 9 and height 21. So scale factor is (scaled length)/(original length) = 9/6 = 3/2 (for width) or 21/14 = 3/2 (for height). So that's the scale factor.

Step2: Calculate scale factor

Take corresponding sides: original width (left) is 6, scaled width (right) is 9. So scale factor = 9/6 = 3/2. Or original height (left) is 14, scaled height (right) is 21. 21/14 = 3/2. So scale factor is 3/2.

Answer:

$\frac{3}{2}$