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Math Expert V
Joined: 02 Sep 2009
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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% (02:01) correct 32% (02:16) wrong based on 178 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 27386 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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7
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..

shaded : unshaded is 5:3..

IMO option D.
Attachments Square.png [ 19.41 KiB | Viewed 20831 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

Shaded area : Unshaded area = 5:3

Answer (D)
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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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Answer is D. Here's why:

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

Area shaded/area unshaded = 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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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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1
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?
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
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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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_________________ Re: ABCD is a square. E and F are the midpoints of sides CD and BC, respec   [#permalink] 07 Apr 2019, 20:45
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# ABCD is a square. E and F are the midpoints of sides CD and BC, respec

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