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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # ABCD is a square. E and F are the midpoints of sides CD and BC, respec

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Math Expert V
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ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 68% (02:02) correct 32% (02:00) wrong based on 254 sessions

### HideShow timer Statistics 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 the unshaded region?

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

Attachment: T7405.png [ 11.92 KiB | Viewed 32917 times ]

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Bunuel wrote: 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 the unshaded region?

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

Attachment:
The attachment T7405.png is no longer available

Let's join the midpoints as show in the figure then we'll get four small squares...

AODX be 1 and similarly all squares..and we can split the unit 1 into half and half as per the mid points...

Then shaded will be 1+1/2+1/2+1/2 = 5/2

and unshaded is 1/2+1/2+1/2 = 3/2..

IMO option D.
Attachments Square.png [ 19.41 KiB | Viewed 25228 times ]

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Let side of the square be 2.

Area of upper shaded region = 1/2 * 2 * 2 = 2
Area of lower shaded region = 1/2 * 1 * 1 =0.5

Total area of shaded region = 2+0.5=2.5

Area of unshaded region = Area of square - Area of shaded region = 2*2 - 2.5 = 1.5

Ratio = 2.5:1.5 = 5:3

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Bunuel wrote: 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 the unshaded region?

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

Attachment:
T7405.png

BD splits the square into two equal triangles. So area of triangle ABD is half the area of square ABCD.

Since C and E are mid points of sides, triangles CEF and CDB are similar with sides in the ratio 1/2. So the areas of the two triangles will be in the ratio (1/2)^2 = 1/4.
Area of CEF is 1/4 the area of triangle CDB which is half the area of square.

Total shaded area = (1/2)*area of square + (1/4)*(1/2) * area of square = 5/8 * area of square
Total unshaded area = 3/8 * area of square

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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D 5:3? I did not do any calculations, since I got a similar q in a CAT

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Say s = 4

Area shaded region = 8 + [(1/2)(2)(2)] = 10 (half area of square + area of triangle)
Area unshaded region = total area - area shaded = 16-10 = 6

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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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suppose side is equal to 4.
total area 16
half is 8.

now...since E and F are mid-points, triangle EFC is a 45-45-90 triangle, in which EC and FC are legs. We know the length of the legs, we can find the area.
area is 2*2/2 = 2.
shaded area is 8+2 = 10
unshaded area is 16-10 = 6
10:6 = 5:3
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Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Top Contributor
Bunuel wrote: 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 the unshaded region?

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

Attachment:
T7405.png

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?
We can create an EQUIVALENT ratio by multiplying top and bottom by 2 to get: 5/3, which is the same as 5 : 3

Cheers,
Brent
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ABCD is a square. E and F are the midpoints of sides CD and BC, respec  [#permalink]

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Bunuel wrote: 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 the unshaded region?

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

Attachment:
T7405.png

This is how i solved ... save way as Brent from GMATprepnow who is my teacher and he is awesome but with my example Let's say the sides of the square have length 4.

So, ∆ABD is a triangle with a base of length 4 and a height of length 4
So, area of ∆ABD = 1/2*4*4 = 8

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 2 and a height of length 2
So, area of ∆EFC = 1/2*2*2= 2

So, the TOTAL area of the 2 shaded regions = 8+ 2 = 10

Since the area of the SQUARE = (4)*(4) =16, and since the TOTAL area of the 2 shaded regions = 10, we can conclude that the area of the UNSHADED region = 16-10 = 6

What is the ratio of the shaded region area to unshaded region?
We can create an EQUIVALENT ratio by multiplying top and bottom by 2 to get: 5/3, which is the same as 5 : 3 ABCD is a square. E and F are the midpoints of sides CD and BC, respec   [#permalink] 20 Nov 2019, 10:40

# ABCD is a square. E and F are the midpoints of sides CD and BC, respec  