Another one from Mind Your Decisions that YT suggested I watch a week or two ago:
He then does it the obvious and quick but rather boring way, using Pythag.
I watched the first minute (so that I didn't know his method or the answer) and then decided to drift off to sleep to see whether my subconscious could work out how to do it using the 'co-ordinates' method. At four in the morning, I dreamed that I knew how to do it and was explaining it to my daughter (who's doing a maths degree). So I woke up and tried it; my subconscious had cracked it for me!
First, redraw the picture, mentally at least:
The formula for a circle is X^2 + Y^2 = 1.
The two blue outlined squares touch the origin (0,0) and point D.
The equation for the green line is Y = X/2.
Point D has co-ordinates that satisfy both equations.
So (X^2)/4 + X^2 = 1.
Point D has co-ordinates X = √0.8, Y = √0.2.
Point B has co-ordinates (0,√0.2).
Length AB = √0.2.
The orange triangle must be a 90-45-45 triangle, so length AC also = √0.2
Length BC (using Pythag) = √0.4.
BC is the side length of the square, so the area of the square = 0.4.
Debate Magazine
