Question 238717: ABCD is a rectangle, and its length is twice its width. The perimeter of the rectangle is 60 units. P, Q, R, and S are the midpoints of each side. What is the area of the quadrilateral PQRS?
Answer by JimboP1977(311)   (Show Source):
You can put this solution on YOUR website! The old saying that a picture says a thousand words is no more appropriate than here!
First thing to do is draw the rectangle out with the mid points.
A______P______B
|**************|
|**************|
S**************Q
|**************|
|**************|
D______R______C
Then let's put down what we know about this rectangle:
AB = 2AD (1)
2AB+2AD = 60 (2)
Subsituting in AB from (1) into (2) gives
2AB+AB = 60
3AB = 60
So we know AB = 20 units
So as we know P is halfway between A and B we know that AP = 10 units
S is halfway between AD so AS must be 5 units
We know that all the corner triangles APS, BPQ, CQR, DRS are equal in size so
The total area of the rectangle ABCD minus APS, BPQ, CQR and DRS must equal the area of the remaining quadrilateral PQRS.
Area of APS = 5*10*1/2 = 25 units
So the four triangles must have an area of 4*25 = 100
Rectangles area = AB*AD = 20*10 = 200
200 - 100 = 100 units
So PQRS = 100 units.
Does this make sense? I would suggest you draw it out for youself and go through each step to make sure you understand it.
James
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