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Points P, Q, R and S divide the respective sides of rectangle ABCD in the proportion 1:2. If the area of rectangle ABCD is 1, find the area quadrilateral PQRS

 

 May 14, 2020
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Since P divides sides AB into a ratio of 1:2, if AP = x, then PB = 2x.  Also, CR = x and RD = 2x.

 

Since S divides sides DA into a ratio of 1:2, if DS = y, then SA = 2y.  Also, QB = y and QC = 2y.

 

The area of triangle(APS)  =  ½·AP·SA  =  ½·x·2y  =  xy.

The area of triangle(RCQ)  =  ½·RC·CQ  =  ½·x·2y  =  xy.

 

The area of triangle(PBQ)  =  ½·BP·BQ  =  ½·2x·y  =  xy.

The area of triangle(RDS)  =  ½·RD·DS  =  ½·2x·y  =  xy.

 

So, the total area of these 4 triangles is 4xy.

 

The area of the rectangle ABCD  =  3x·3y  =  9xy.

 

So, the area of PQRS  =  9xy - 4xy  =  5xy.

 

The ratio of the area of PQRS to ABCD  =  5xy : 9xy  =  5 :9.

 

If the area of ABCD = 1, then the area of PQRS = 5/9.

 May 14, 2020

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