Question

preimage image scale factor = 9 ? 135 in a = 14,580 in² given the area, what is the length of the missing side in the preimage? 12 in 24 in 3 in 15 in

Explanation:

Step1: Find side of image square

The image is a square (since area formula \(A = s^2\) and we can check: \(135\times135 = 18225\)? Wait, no, wait the area is 14580. Wait, maybe the image is a rectangle? Wait, no, the preimage and image are similar, scale factor 9. Wait, the image has one side 135 in, scale factor 9, so preimage side would be \(135\div9 = 15\) in? Wait, let's check area. Image area: if it's a rectangle, but maybe it's a square? Wait, no, the area of image is 14580, and one side is 135, so the other side of image is \(14580\div135 = 108\)? No, that doesn't make sense. Wait, maybe both are squares. Wait, preimage is a square, image is a square with scale factor 9. So side of image: let's see, if preimage side is \(x\), image side is \(9x\). Area of image is \((9x)^2=81x^2\). But also, image has a side 135, so if it's a square, \(9x = 135\), so \(x = 135\div9 = 15\). Let's check area: preimage area would be \(15^2 = 225\), image area \(135^2 = 18225\), but the given image area is 14580. Wait, maybe it's a rectangle. Wait, image has length 135, area 14580, so width of image is \(14580\div135 = 108\). Then preimage length is \(135\div9 = 15\), preimage width is \(108\div9 = 12\). But the question is about the missing side in preimage. Wait, the preimage is a rectangle, one side is missing? Wait, the preimage is a square? No, the diagram shows preimage as a square? Wait, maybe the image is a rectangle with length 135 and area 14580, so width is 108. Then preimage, scale factor 9, so preimage length is \(135\div9 = 15\), preimage width is \(108\div9 = 12\). But the question is "the length of the missing side in the preimage". Wait, maybe the preimage is a square? No, maybe I misread. Wait, the scale factor is 9, so the image is 9 times the preimage. So if the image has a side 135, then preimage side is \(135\div9 = 15\). Let's check: preimage side 15, image side 135 (159). Then area of image would be 135 (15*9)? No, wait, no, if it's a square, preimage side x, image side 9x, area (9x)^2 = 81x². But given image area is 14580, so 81x²=14580 → x²=14580/81=180 → x=√180≈13.4, which is not 15. Wait, maybe it's a rectangle, image has length 135, width w, area 135w=14580 → w=14580/135=108. Then preimage length is 135/9=15, preimage width is 108/9=12. So the missing side in preimage: if preimage is a rectangle, one side is 12? No, the options are 12,24,3,15. Wait, the preimage is a square? No, maybe the diagram shows preimage as a square with missing side, and image as a square with side 135. Wait, 135 divided by 9 is 15. So the missing side is 15. So the correct answer is 15 in.

Answer:

15 in