Quadrilateral ABCD in the figure below is a square and CE = DE.
Find the area ratio of the square ABCD and the triangular BFG of shadow area.
Find the area ratio of the square ABCD and the triangular BFG of shadow area.
Answer
The area of ABCD is set to 3 × 4 = 12.
The ratio to be found is 12 : 1.
12 : 1
Solution
Since it is DF = BF, BF : BD = 1 : 2.
Since △ABG and △CEG are homothetic and homothetic ratio is 2 : 1, BG : BE = 2 : 3.
If the area of △BFG of a shadow area is set to 1, the area of △BDE will be set to 1 × 2/1 × 3/2 = 3.
Since the area of △BCD is twice the area of △BDE and the area of a square ABCD is twice the △BCD, the area of a square ABCD is the 2 × 2 = 4 times of △BDE.
Since △ABG and △CEG are homothetic and homothetic ratio is 2 : 1, BG : BE = 2 : 3.
If the area of △BFG of a shadow area is set to 1, the area of △BDE will be set to 1 × 2/1 × 3/2 = 3.
Since the area of △BCD is twice the area of △BDE and the area of a square ABCD is twice the △BCD, the area of a square ABCD is the 2 × 2 = 4 times of △BDE.
The area of ABCD is set to 3 × 4 = 12.
The ratio to be found is 12 : 1.
