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Why the Golden Ratio Appears Between Two Ordinary Straight Lines
A 2021 AMC problem begins with two simple slopes and ends at one of mathematics’ most famous numbers. Draw two straight lines through the origin:
Now draw a third line exactly halfway between them in angle. Suppose its equation is
What is k?
There is an answer that almost asks to be chosen: 2.
After all, the first line has slope 1, the second has slope 3, and 2 sits exactly halfway between them.
Unfortunately, geometry does not work that way.
The correct answer is
the number better known as the golden ratio.
Its appearance is surprising because there are no pentagons, Fibonacci numbers, golden rectangles, or obvious references to the golden ratio anywhere in the problem. There are only two ordinary straight lines.
So where does √5 come from?
That is what makes this problem worth examining.
A problem from the AMC 12
The question appeared as Problem 13 of the Fall 2021 AMC 12A, part of the American Mathematics Competitions program run by the Mathematical Association of America. The AMC 12 is a 25-question, 75-minute competition for students in…
