MathDB

The diagram shows a

20

20

square

ABCD

. 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

EJF

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.
[asy] size(175); defaultpen(linewidth(0.8)); real r=20/8; pair x[]; draw(origin--(0,20)--(20,20)--(20,0)--cycle); string label[]={"B","K","E","J","F","I","G","H","C"}; for(int i=1;i<=7;i=i+1) { if(floor(i/2)==i/2) { x=(i*r,0); label("

"+label+"

",x,S); } else { x=(i*r,20); label("

"+label+"

",x,N); } } filldraw(origin--x[1]--x[2]--x[3]--x[4]--x[5]--x[6]--x[7]--(20,0)--(20,10)--(0,10)--cycle,gray); label("

B

",origin,SW); label("

C

",(20,0),SE); label("

A

",(0,20),NW); label("

D

",(20,20),NE); label("

M

",(20,10),E); label("

L

",(0,10),W); [/asy]

Problem Statement