Are all equilateral triangles are similar? - Answers

DFN: we call a triangle equilateral if all sides of the triangle are the same length DFN:we call two triangles similar if corresponding angles are equal, and corresponding sides are proportional. First show that all corresponding sides are proportional: Consider a equilateral triangle with side lengths 1, all other equal lateral triangles sides can be expressed as S*(1), where S is some scalar. Hence all equilateral triangles sides are proportional to each other. Next, show that all corresponding angles are equal: The angle between two sides of a triangle is related to the length of the sides. These relationships are called sin, cos, and tan. Knowing that the cos(x), where x is one of the angles in the triangle, is the adjacent divided by the hypotenuse we see that cos(x)=(1/2)c/a, since a = c (because its equal lateral) we are left with cos(x)=(1/2) which means x = 60 degrees. this can be applied to all three angles, which shows that all three angles are 60 degrees. / \ / | \ a / | \ b /__ |__\ c We have now shown that all equal lateral triangles are similar because they all have proportional sides, and they all have equal angles.