In unit square, ABCD, A is joined to the midpoint of BC, B is joined to the midpoint of CD, C is joined to the midpoint of DA, and D is joined to the midpoint of AB.
Find the area of the shaded square formed by this construction.
We shall solve this in two different ways. First the hard way... (c;
Using the Pythagorean Theorem, BE2 = 12 + (1/2)2 BE = 5/2.
As ΔBEC is similar to ΔFEC,
BE/BC = CE/FC, 5/2 = (1/2)/FC FC = 1/5.
FE/CE = CE/BE, FE/(1/2) = (1/2)/(5/2) FE = 1/(25).
HC = HG + GF + FC, and as HG = FE, 5/2 = 1/(25) + GF + 1/5.
5/(25) = 3/(25) + GF GF = 1/5.
Hence the area of the shaded square = 1/5.
Now the easy way...
It can be seen that ΔEFC is congruent with ΔEID.
Area five squares = Area of unit square, ABCD = 1
Area of one square = Area of the shaded square = 1/5
What about the area of the shaded octagon in the diagram below?
Is it a regular octagon?