ABCD is a square, E and F are the midpoints of sides...

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ABCD is a square, E and F are the midpoints of sides CD and BC, respectively. What is the ratio of the shaded region area to unshaded region?

A. 1 : 1
B. 2 : 1
C. 3 : 1
D. 5 : 3
E. 8 : 3

The OA is D.

I know that the area of the square is L^2, and the area of the shaded region will be the sum of areas of the two triangles,

The area of the bigest triangle will be,

1/2* L^2, because b = h

Then, the area of the smallest triangle will be,

1/2*(L^2)/4

The area of the shaded region will be,

1/2*L^2 + 1/8*L^2 = 5/8*L^2

The area of unshaded region will be

L^2 - 5/8*L^2 = 3/8*L^2

Finally, Shaded region / unshaded region will be,

$$\frac{5/8*L^2\ }{3/8*L^2}=\frac{5}{3}$$

Is there a strategic approach to this question? Because I think that this is the long way to solve it. Can any experts help? Thanks!

[Brent@GMATPrepNow]

ABCD is a square, E and F are the midpoints of sides CD and BC, respectively. What is the ratio of the shaded region area to unshaded region?

A. 1 : 1
B. 2 : 1
C. 3 : 1
D. 5 : 3
E. 8 : 3

The OA is D.

I know that the area of the square is L^2, and the area of the shaded region will be the sum of areas of the two triangles,

The area of the bigest triangle will be,

1/2* L^2, because b = h

Then, the area of the smallest triangle will be,

1/2*(L^2)/4

The area of the shaded region will be,

1/2*L^2 + 1/8*L^2 = 5/8*L^2

The area of unshaded region will be

L^2 - 5/8*L^2 = 3/8*L^2

Finally, Shaded region / unshaded region will be,

$$\frac{5/8*L^2\ }{3/8*L^2}=\frac{5}{3}$$

Is there a strategic approach to this question? Because I think that this is the long way to solve it. Can any experts help? Thanks!

Your approach is perfect, but it's a bit time-consuming.
Another approach is to assign some nice values to the diagram.

Cheers,
Brent

[Brent@GMATPrepNow]

ABCD is a square, E and F are the midpoints of sides CD and BC, respectively. What is the ratio of the shaded region area to unshaded region?

A. 1 : 1
B. 2 : 1
C. 3 : 1
D. 5 : 3
E. 8 : 3

Another approach is to assign some nice values to the diagram.

Let's say the sides of the square have length 2.

So, ∆ABD is a right triangle with a base of length 2 and a height of length 2
So, area of ∆ABD = (2)(2)/2 = 2

Since , E and F are the midpoints of sides CD and BC, respectively, we know that ∆EFC is a right triangle with a base of length 1 and a height of length 1
So, area of ∆EFC = (1)(1)/2 = 0.5

So, the TOTAL area of the 2 shaded regions = 2 + 0.5 = 2.5

Since the area of the SQUARE = (2)(2) =4, and since the TOTAL area of the 2 shaded regions = 2.5, we can conclude that the area of the UNSHADED region = 4 - 2.5 = 1.5

What is the ratio of the shaded region area to unshaded region?
area of shaded region area/area of unshaded region = 2.5/1.5
We can create an EQUIVALENT ratio by multiplying top and bottom by 2 to get: 5/3, which is the same as 5 : 3

Answer: D

Cheers,
Brent