The diagram shows a 20 by 20 square . The points E, F and G are equally spaced on side BC. The points H, I, J, and K on side DA are placed so that the triangles BKE, EGF, FIG, and GHC are isosceles. Points L and M are midpoints of the sides AB and CD, respectively. Find the total area of the shaded regions.
fudge I accidently reloaded my page 2 times already, meaning that I had to re do my explanation two times.
calculate KBE
Base:
20/4 = 5
Height:
20
5 * 20 = 100 / 2 = 50
base of grey triangle is half of KBE, because BC is divided into 4 congruent parts, while LM, equal to BC, is divided into 8 parts.
So calculate grey triangle:
Base:
5/2 = 2.5
Height:
20 / 2 = 10
10 * 2.5 = 25/2 = 12.5
Count number of grey triangles. 8 triangles
So 12.5 * 8 = 100
I get the same as CU with a little different reasoning
BE = (1/2) BC = 20/4 = 5
And the height of triangle BKE = 20
So...it's area = (1/2) (5) (20) = 50
The gray portion of this triangle has 1/2 the base and /2 the height of BKE
So its area = (1/2) (1/2) (50) =50/4
And we have 8 of these congruent gray areas....so....the total gray area = 8 (50/4) = 100 units^2