Problem Solving

A square ABCD is drawn and point E is marked on AB such that AE=AB/3. Similarly points F, G and H are marked on the sides of the square such than BF=BC/3, CG=DE/3 and DH=DA/3. If the points E, F, G and H are connected to make a parallelogram, what is the ratio of the area of square ABCD to the area of parallelogram EFGH?

A. 81/16
B. 9/4
C. 9/5
D. 5/4
E. 4/5

The OA is C.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

swerve wrote:A square ABCD is drawn and point E is marked on AB such that AE=AB/3. Similarly points F, G and H are marked on the sides of the square such than BF=BC/3, CG=DE/3 and DH=DA/3. If the points E, F, G and H are connected to make a parallelogram, what is the ratio of the area of square ABCD to the area of parallelogram EFGH?

A. 81/16
B. 9/4
C. 9/5
D. 5/4
E. 4/5

The OA is C.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

Let AB = BC = CD = DA = 3.
So AE = BF = CG = DH = 1 and EB = FC = GD = HA = 2.

Hence, Area of triangle AEH = Area of triangle EBF = Area of triangle FCG = Area of triangle GDH = 1/2 * *length * height = 1/2 (2 *1) = 1

Area of all 4 triangles = 4.

Area of square ABCD = AB^2 = 3^2 = 9.

Area of parallelogram EFGH = Area of the square ABCD - the area of all 4 triangles.
Hence, Area of parallelogram EFGH = 9 - 4 = 5.

Hence the ratio of ABCD to EFGH = 9:5.

C