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
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.