Similar figures is shape in geometry that have the same shape but not always the same size.[1][2][3] This means that one figure can look
Similar figures is shape in geometry that have the same shape but not always the same size.[1][2][3] This means that one figure can look like another figure, like a smaller or bigger version, but the angles still match and the shape stays the same. In geometry, similarity is closely connected to dilations, which are transformations that change the size of a figure but keep its angle measures the same. A dilation always preserves angle measure, but it does not always preserve distance, meaning the lengths of sides can change unless the scale factor is 1. When the scale factor is 1, the image is congruent to the preimage, meaning they are exactly the same size and shape. Similar figures are created when a figure is transformed in a way that keeps all corresponding angles congruent and all corresponding side lengths in proportion.[1] “In proportion” means the sides are in the same ratio, so one figure may have sides that are all multiplied by the same number compared to the other figure. For example, if one figure has sides that are twice as long as another figure, then the scale factor is 2, and the figures are similar. Similar figures are important because they allow us to compare shapes even when they are different sizes.[1] Corresponding angles of similar figures are always congruent, which means they have the same measure, and corresponding sides of similar figures are proportional, which means they keep the same relationship in size. This is why similar figures look the same even if one is bigger or smaller.[2] Similarity is often created by rigid motions combined with dilations. Rigid motions preserve distance and angle measures, so they create congruent figures, while dilations change size but keep shape the same, creating similarity.[2] A key idea connected to similar figures is the AA similarity theorem, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Another important idea is the SAS similarity theorem, which states that if two sides are in proportion and the included angle is congruent, then the triangles are similar. Similar figures are also connected to other geometry ideas like the side splitter theorem, which shows that when a line is parallel to one side of a triangle, it divides the other two sides proportionally. In this way, similar figures help explain how shapes can stay the same in form even when they are resized. The perimeter of a similar figure changes by multiplying by the scale factor, while the area changes by multiplying by the square of the scale factor, written as k², where k is the scale factor. This shows that similarity affects different parts of a shape in different ways.
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