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3 Responses
Think of your rectangle as a photo trying to slip into a picture frame without bending its corners. To “fit” (no cropping), you compare how much you’re allowed to stretch by width and by height, then pick the smaller stretch so neither side overshoots. In symbols: scale-to-fit = min(target width / original width, target height / original height). If you wanted to “fill” (cover the whole frame, with cropping allowed), you’d pick the larger ratio instead.
Worked example with your numbers: width ratio = 10/4 = 2.5, height ratio = 12/7 ≈ 1.714…, so for a proper fit you take the smaller one: 12/7. Scale the original by 12/7 to get 4 × (12/7) ≈ 6.857 wide and 7 × (12/7) = 12 tall-snug on height, comfy margin on the sides. If you chose the larger ratio (2.5), you’d get 10 by 17.5, which fills the width but overflows the height (crop alert!). For a quick extra: fitting a 5×5 into 8×12 uses min(8/5, 12/5) = 1.6, giving 8×8. For more on scale factors and keeping shapes proportional, see Khan Academy’s intro: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:expressions-equations/x2f8bb11595b61c86:scale-factors/v/introduction-to-scale-factor
Neat rule: to make it fit without cropping, scale by the smaller container/object side ratio-min(10/4, 12/7) = min(2.5, 1.714…) = 1.714…, giving about 6.86×12; to make it fill (allowing crop), use the larger ratio, here 2.5. Like sliding a photo into a frame-stop when the first edge touches for fit, keep zooming until both sides cover for fill.
You’ve got the right idea-when you want the resized rectangle to fit entirely inside the box without cropping (often called “contain”), you take the smaller of the two ratios: box width/original width and box height/original height. That guarantees both dimensions stay within the box. For 4×7 into 10×12, compare 10/4 = 2.5 and 12/7 ≈ 1.714; the smaller is 12/7, so scale by about 1.714. That gives a new size of about 6.857×12, which fits with room to spare on the sides. If instead you wanted to completely fill the 10×12 box (possibly cropping), you’d use the larger ratio, 2.5, getting 10×17.5 and trimming the excess height. It’s just like slipping a photo into a frame: “fit” keeps the whole picture with some matting, while “fill” covers the frame but may chop off the edges. Nice visual walkthroughs of scale factors and dilations are here: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-dilations/a/what-is-a-dilation