Kinesys touches down on Polygon with a clatter of sharp edges. Everything here is made of polygons and solids: the houses are cubes, the trees are pyramids, and even the clouds are octahedra drifting placidly across the sky. Curves are considered an eccentric affectation, tolerated only in the arts district.
And right there, in the middle of a square, is the scandal of the day: a perfectly circular fountain with an expertly drawn rectangle inside it. The Polygonians are arguing heatedly about how long its diagonal is.
For once, Dord agrees with the locals: “A circle! The nerve.” Liz just wants to find a spare part. M00N, who is round, feels a little self-conscious.
The Riddle
A rectangle ABCD is inscribed in a quarter circle, as shown in the figure: vertex B is at the center of the circle, sides BA and BC lie along the two radii that bound the quarter circle, and vertex D lies on the circle.
Find the exact length of diagonal AC, given that the circle has a radius of 1.
Hint
A rectangle has two diagonals, but they aren’t equally convenient: one of them can be measured without any calculation at all.
Solution
AC is 1 unit long.
AC is one diagonal of the rectangle; the other diagonal, BD, runs from the center of the circle to a point on the circle, so it is a radius and measures 1. Since the two diagonals of a rectangle are equal, AC is also 1. No calculation needed: you just have to look at the “right” diagonal.