Suppose we have a square ABCD and we draw a circumference which goes through point A and the middle points of the sides BC and CD.

Which has the biggest perimeter, the square or the circle?

Guest Feb 6, 2021

#1**0 **

An easy way is to look at it as a co-ordinate geometry problem.

Suppose that A is the origin, (with AB as the x-axis), and that the square has a side length of 2.

The point on the circle at the midpoint of DC, call it E, will have co-ordinates (1, 2).

The centre of the circle, call it F, will be on the line AC, suppose that it has co-ordinates (k, k).

Then, if the radius of the circle is r,

AF^2 = FE^2 = r^2,

and using Pythagoras,

k^2 + k^2 = (1 - k)^2 + (2 - k )^2,

2k^2 = 1 - 2k + k^2 + 4 - 4k + k^2,

6k = 5,

k = 5/6.

r^2 = k^2 + k^2,

etc..

Guest Feb 7, 2021