Trigonometry Identities

Consider a collection of ﬁve points evenly spaced around a circle to form a regular pentagon. Assume the ﬁgure is scaled so that the sides of the pentagon have length 1. Question: Use Ptolemy’s theorem to calculate the distance between non-adjacent vertices.

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Let the vertices of the pentagon, in counterclockwise order, be $A,B,C,D,E$. Consider the cyclic quadrilateral $ABCD$. Let $x=AC=BD=DA$. By Ptolemy's Theorem, we have $(x)(x)=(1)(1)+(x)(1)$. Solve the quadratic equation, using the Quadratic Formula.