In the figure, ABCD is a square of side length 1, and M and N are the midpoints of sides AB and CD, respectively. Find the area of the shaded region.

Guest May 16, 2017

#1**+2 **

MBND is a parallelogram with a base = 1/2 and height = 1

So....its area = (1/2) (1) = 1/2

And....because of symmetry, the diagonal AC divides the parallelogram into equal areas......so.....the shaded area = 1/4

CPhill
May 17, 2017

#2**+1 **

DMBN is a parallelogram that is cut in half by AC.

area of shaded region = (1/2)(DMBN)

Imagine that DMBN is a stack of saltine crackers that is leaning to the right.

If we straightened the crackers up, the area of the side facing us stays the same.

When they are straightened up, we can see that

area of side facing us = MB * MN

And... MN = 1 and MB = (1/2)

So..

area of DMBN = (1)(1/2) = (1/2)

and so..

area of shaded region = (1/2)(1/2) = 1/4 un^{2}

hectictar
May 17, 2017