Dilation: Lengths Scale, Angles Stay

Picture a photocopier set to 150 percent. Every segment in the image gets longer, but the corners look just as sharp as before. That's dilation: a transformation that multiplies every length by a scale factor k, while leaving every angle measure exactly as it was. Nothing rotates or skews, so the shape only grows or shrinks uniformly from a center point.

The mechanism is about what an angle actually measures. An angle records a turn between two rays, not a distance along them. When you dilate, both rays get longer by the same factor k, but the amount they've rotated away from each other never changes. Stretch two rays from a vertex by the same amount and the gap between them, measured in degrees, is untouched. That is also why dilated figures are always similar to the original: same shape, same angles, just a different size, with corresponding sides in a fixed ratio of k.

This idea connects directly to triangle problems involving similarity and right-triangle trig. Since angles survive dilation, the sine, cosine, and tangent of any angle in the figure survive too, which is why scaling a right triangle never changes its trig ratios. On questions about area and perimeter, remember that length scales by k but area scales by k², since area is a product of two lengths.