What are Similar Figures?:
Figures are mathematically similar when all corresponding angles are congruent, they are related by a scale factor, and they have the same or relating ratios. Corresponding angles are angles that are in the same place on each figure. A scale factor is the number that two figures are related by. For example, if there is a 2×4 rectangle (rectangle a) and a 4×8 rectangle (rectangle b) the scale factor from a to b would be two because 2×2 is 4. Figures being mathematically similar means that they can be enlarged and shrunk in order to create each other.
True or False:
Any two rectangles are similar: FALSE! All rectangles have congruent corresponding angles because they have four 90º angles. Even though they have congruent corresponding angles, not all rectangles are related by a scale factor. For example, if there was a 1×4 rectangle and 1×6 rectangle they wouldn’t be similar. They would both have 90º angles, but they wouldn’t be related by a scale factor. 1×1=1 and 4×1.5=6. 1 and 1.5 aren’t the same number. Also, they wouldn’t have the same or related side length ratios.
Any two equilateral triangles are similar: TRUE! When two triangles have congruent corresponding angles they are automatically similar. Equilateral triangles have three 60º angles. Also, equilateral triangles have the same side lengths. This means that there will always be a scale factor because all the sides have to be equal. For example, if you have a equilateral triangle with the the perimeter of 9 units (each side is 3 units) and a triangle with the perimeter of 12 units (each side is 4 units), they will automatically have a scale factor. The scale factor from the first triangle to the second one is 1 1/3 and the scale factor from the second one to the first one is 3/4. Another way you can see that these two triangles are similar is by side length ratios. 3/3 and 4/4 are the side length ratios within each triangle. Now you just make sure it is a true proportion by multiplying 3×4 and 3×4. These both equal 12. This shows that these two triangles are similar.