Ampliação E Redução De Figuras Planas

Scaling Plane Figures Without Losing Your Mind

The most basic way to enlarge or reduce a plane figure is through coordinate scaling. You take each vertex of the figure, multiply its x and y coordinates by the scale factor, and then connect the new points. That's really all there is to it mathematically. But the way you set up those coordinates before you even start calculating determines whether your final drawing looks clean or completely distorted. ampliação e redução de figuras planas isn't actually two different topics. It's the same operation with a scale factor greater than one versus less than one. The formula doesn't change. If you have a point P(x, y) and a scale factor k, the new point becomes P'(kx, ky). When k = 2, you're enlarging. When k = 1/3, you're reducing. The mechanics are identical.

Here's where most people fumble. They pick a random point on the figure and call it the center of homothety, then multiply everything outward from there. That works fine on paper with a triangle, but the moment you try this with something like a floor plan or a technical drawing with dozens of vertices, the small rounding errors at each coordinate multiply into visible distortion. I once spent forty-five minutes redrawing a scaled floor plan because the original author had chosen a corner point as the center and then rounded each coordinate to one decimal place. The walls didn't meet. Not even close. The fix was simple: I repositioned the center of homothety to the origin (0, 0) and recalculated every point using fractions instead of decimals until the very end. That cut my correction time from an hour down to about eight minutes. The deeper issue that nobody explains well is the area relationship. When you apply a scale factor k to a figure, the perimeter scales by k, but the area scales by k squared. So if you double every linear dimension, the area becomes four times larger, not two. Students routinely miss this because they're focused on getting the coordinates right and forget to check whether the resulting area makes sense as a sanity test. A triangle that should visually look twice as big in every direction but ends up with half the area? That's your signal that you used k = 1/2 somewhere instead of k = 2, or that you forgot to square the factor when comparing areas.

Another thing worth noting: homothety preserves angles absolutely. It does not preserve distances between arbitrary points on the original figure unless those distances happen to lie along rays from the center of homothety. This matters when you're working with irregular polygons where you need to verify that a diagonal or an internal line segment scales correctly. If you're manually checking your work, measure one diagonal before and after. If it doesn't scale by exactly k, you've made an error somewhere in the coordinate multiplication. The main limitation of this method is that it assumes you know the exact coordinates of every vertex. In practice, that's fine for textbook problems and digital work, but on a construction site or in a drafting office where you're given a physical drawing and a scale ratio, you're often working from measurements taken with a ruler or a scale bar. In those cases, measurement error becomes the bottleneck, not the math. A single millimeter of error on a large drawing can throw off multiple vertices. The workaround I use is to anchor your scaling to at least two known reference points that are far apart on the figure, then derive all other coordinates relative to those. This constrains the error and makes it much easier to spot when something has gone wrong.

For digital work, using a vector graphics program or even a spreadsheet with coordinate columns is dramatically faster than manual calculation. You set up two columns for the original x and y values, multiply each by k in an adjacent column, and plot the results. What used to take ten minutes by hand takes about thirty seconds this way. The approach breaks down completely when you need to scale a figure that isn't defined by discrete vertices — like a freeform curve or a scan of a hand-drawn shape. In that case you're looking at image scaling algorithms like bilinear or bicubic interpolation, which are a different problem entirely and introduce their own artifacts like blurring or pixelation that don't exist in the pure geometric method.