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# In the figure, ABCD is a square, and all the dots are evenly spaced:

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Math Expert
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In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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29 Oct 2017, 05:25
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Question Stats:

86% (01:59) correct 14% (01:46) wrong based on 89 sessions

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In the figure, ABCD is a square, and all the dots are evenly spaced: each vertical or horizontal distance between two adjacent dots is 3 units. Find the area of the shaded region.

A. 60
B. 72
C. 81
D. 96
E. 120

Attachment:

ldvagm_img1.jpg [ 11.56 KiB | Viewed 1958 times ]

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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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29 Oct 2017, 05:49
72.. just rearrange the diag to get half the square

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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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29 Oct 2017, 19:15
B 72
There are six parallelograms with an area of 9 units square each.
There are four right angled triangles with an area of 4.5 units square each

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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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29 Oct 2017, 19:28
Rearranging blue areas we see that from every single of the 16 sqaures a half portion is coloured. So total coloured sqaures are 8 , hence 9*8 = 72

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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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30 Oct 2017, 10:46
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Bunuel wrote:

In the figure, ABCD is a square, and all the dots are evenly spaced: each vertical or horizontal distance between two adjacent dots is 3 units. Find the area of the shaded region.

A. 60
B. 72
C. 81
D. 96
E. 120

Attachment:
The attachment ldvagm_img1.jpg is no longer available

Attachment:

ccccc.jpg [ 26 KiB | Viewed 1534 times ]

The dots are evenly spaced; distances between dots are equal.
As long as the shapes' side lengths are made with the same number of dots, they are congruent.

"Move" some of the shaded parts, see diagram, left square.

Right hand square: Now the shaded area is half the square's area.

One side of the square = 4 dots * 3 units = 12
Square area = 12$$^2$$ = 144

Shaded area = half of square = 72

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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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31 Oct 2017, 12:59
There are 16 triangles of area 4.5
16*4.5 =72
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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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31 Oct 2017, 13:32
I got 72, which is B. All I did was count the triangles. There are probably more efficient ways.
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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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01 Nov 2017, 11:20
IMO B

Total area of the square is (3*4)^2= 144
If we counted the number of blue triangles on the left hand-side, we will realize they complete the blue ones on the left hand-side to make half of the total area of the square. Therefore, the shaded area is 144/2 = 72
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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:  [#permalink]

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10 Sep 2018, 08:29
Top Contributor
Bunuel wrote:

In the figure, ABCD is a square, and all the dots are evenly spaced: each vertical or horizontal distance between two adjacent dots is 3 units. Find the area of the shaded region.

A. 60
B. 72
C. 81
D. 96
E. 120

Attachment:
ldvagm_img1.jpg

If the vertical or horizontal distance between two adjacent dots is 3 units, then the square has dimensions 12 by 12
So, the area of the ENTIRE SQUARE = (12)(12) = 144

Now, notice that the square is comprised of 32 identical triangles

Since 16 of the triangles are shaded, the shaded regions represent 16/32 (aka 1/2) of the ENTIRE SQUARE

In other words, the area of the shaded region = 1/2 of 144
= 72

Cheers,
Brent
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Re: In the figure, ABCD is a square, and all the dots are evenly spaced:   [#permalink] 10 Sep 2018, 08:29
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