SOLUTION: 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 quadrilat

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

RELATED QUESTIONS
In (SQUARE SYMBOL - rectangle?) ABCD, P, Q, R, and S are mid-points of the sides. The... (answered by ikleyn)

the length of a rectangle is 5 more than twice its width . its perimeter is 70 units.... (answered by JBarnum)

the perimeter of a rectangle is 12 units. the width of the rectangle is 9 units less... (answered by mananth)

A rectangle is twice as long as it is wide. Its perimeter (the sum of the lengths of its... (answered by josmiceli,ikleyn)

the length of a rectangle is 5 units longer than its width. The perimeter of this... (answered by oberobic)

In (square symbol?) ABCD, P, Q, R, and S are mid-points of the sides. The perimeter of... (answered by jim_thompson5910,ikleyn)

In (square symbol?) ABCD, P, Q, R, and S are mid-points of the sides. The perimeter of... (answered by solver91311)

the length of a rectangle is twice its width. if the perimeter is 60, find the dimensions (answered by checkley77)

The length of a rectangle is twice its width. If the perimeter is 60, find the dimensions (answered by ikleyn)