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# mathcounts

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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.

Nov 12, 2019

#1
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here is diagram if needed Nov 12, 2019
#2
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i get what it is saying but i get too many variables that i get confused

Nov 12, 2019
#3
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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

Nov 12, 2019
#4
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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   Nov 12, 2019