Proportions in Geometry and Scaling

A scale drawing is a reduced or enlarged representation of an object. The relationship between the drawing and the real object is called the scale factor. We express this as a ratio: $Scale = \frac{\text{Drawing Length}}{\text{Actual Length}}$. For example, a scale of $1:100$ means that $1$ unit on paper represents $100$ units in real life. To find the actual length, you multiply the drawing length by the denominator of the scale. To find the drawing length, you divide the actual length by that same number. This ensures that every part of the object stays in proportion, meaning the shape doesn't get distorted or stretched.

When you scale a 2D shape, the side lengths change linearly, but the area changes differently. If you double the sides of a square, the area doesn't just double—it quadruples! This is because area is length times width. If the scale factor for length is $k$, the scale factor for the area is $k^2$. For example, if a floor plan has a scale of $1:50$, the actual area is $50^2$ (which is $2,500$) times larger than the area on the paper. Always remember: **Length scales by $k$, Area scales by $k^2$.**

In geometry, two figures are similar if they have the same shape but different sizes. Their corresponding angles are equal, and their corresponding side lengths are proportional. We can use this to find unknown lengths. If Triangle A is similar to Triangle B, the ratio of any two corresponding sides will be equal. We set up the equation: $\frac{\text{Side } A_1}{\text{Side } B_1} = \frac{\text{Side } A_2}{\text{Side } B_2}$. This is the foundation of trigonometry and navigation!