What happens to the area of a figure if we scale it up or down (i.e., enlarge or reduce it)? In Part C, we review the concept of similarity and examine the relationship between a scale factor and the resulting area of the similar figure. Previously we only explored similar triangles, but in this section we will use a variety of shapes.
When we enlarge or reduce a figure, we are using an important mathematical idea: similarity. Similar figures have the same shape but are not necessarily the same size. More formally, we state that two figures are similar if and only if two things are true: (1) The corresponding angles have the same measure, and (2) the corresponding segments are in proportion. Enlarging or reducing a figure produces two figures that are similar. Note 3
The second attribute means that when we are building a similar figure, we must increase or decrease the sides multiplicatively by the scale factor. What happens to the length of each side when we enlarge a figure, say, by a scale factor of 2? Well, since in similar figures the corresponding sides are in proportion, each of the sides of the enlarged similar figure is twice as long as the corresponding side of the original figure. So, for example, in the enlargement of the trapezoid shown below on the left, the enlarged trapezoid is similar to the small trapezoid because the angles are congruent and each of the sides is proportionally larger (twice as long):
Building similar figures, however, is not always so straightforward! For example, the trapezoid below is not similar to the original trapezoid. The angles are congruent, but the corresponding sides are not proportional -- some of the sides have been "stretched" more than others: