Goldilocks and the 3 Similar Triangles
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Geometric objects that have the same shape and same size are called congruent. When they have the same shape but different sizes they are called similar. Here are three similar triangles called Mama Triangle, Papa Triangle, and Baby Triangle. You can change the position of their vertices with the sliders. Since their angles all change by the same amount they remain similar. But notice that even though they are different sizes, the ratio of their areas to the square of ANY linear element is the same for all three triangles (watch the pink line in the table). A teenage Albert Einstein made use of this quality in his proof of the Pythagorean theorem.
Contributed by: John Kiehl (September 2007)
Open content licensed under CC BY-NC-SA
Notice each linear element produces a different, specific number for the ratio of area to linear-element-squared, but in every case all three triangles exhibit that same number.