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# The area of rectangle ABCD is closest to which of the following?

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
The area of rectangle ABCD is closest to which of the following?  [#permalink]

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07 May 2015, 04:05
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Difficulty:

25% (medium)

Question Stats:

68% (01:09) correct 32% (01:16) wrong based on 149 sessions

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The area of rectangle ABCD is closest to which of the following?

(A) 100
(B) 130
(C) 170
(D) 200
(E) 230

Attachment:

drawn-1.png [ 4.01 KiB | Viewed 2532 times ]

Kudos for a correct solution.

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Joined: 25 Nov 2014
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GMAT 1: 670 Q48 V34
The area of rectangle ABCD is closest to which of the following?  [#permalink]

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Updated on: 07 May 2015, 06:56
1
Let length be: L
tan 30 deg = 10/L

Since, tan 30 deg = 1/$$\sqrt{3}$$

1/$$\sqrt{3}$$ = 10/L

L = 10*$$\sqrt{3}$$
($$\sqrt{3}$$ ~ 1.732)
Therefore, L = 17.32

Now, area is 10*L, ie. 10*17.32 = 173.2 ~ 170

Originally posted by ApurbaGMAT on 07 May 2015, 06:49.
Last edited by ApurbaGMAT on 07 May 2015, 06:56, edited 1 time in total.
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Posts: 153
Re: The area of rectangle ABCD is closest to which of the following?  [#permalink]

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07 May 2015, 06:55
1
As we know that Arc ACD is 30 degrees and arc DAC is 90 degrees we can infer that the remaining arc in the upper right area has to be 60 degrees as a triangle has a maximum of 180 degrees.

This yields a 30-60-90 triangle with side relationship of x - x sqrt 3 - 2x. X in this case is 10, therefore we can derive the other sides and also the bottom side of the rectangle.

In my calculation this results in an area of approx. 170!
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Joined: 03 Sep 2014
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Concentration: Marketing, Healthcare
Re: The area of rectangle ABCD is closest to which of the following?  [#permalink]

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07 May 2015, 09:15
1
Bunuel wrote:

The area of rectangle ABCD is closest to which of the following?

(A) 100
(B) 130
(C) 170
(D) 200
(E) 230

Attachment:
drawn-1.png

Kudos for a correct solution.

Triangle ABC or ACD are 30-60-90 triangle, thus, AD= $$a\sqrt{3} = 10\sqrt{3} = 17 => Area = 10*17 = 170$$

Math Expert
Joined: 02 Sep 2009
Posts: 58445
The area of rectangle ABCD is closest to which of the following?  [#permalink]

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11 May 2015, 04:01
Bunuel wrote:

The area of rectangle ABCD is closest to which of the following?

(A) 100
(B) 130
(C) 170
(D) 200
(E) 230

Attachment:
drawn-1.png

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

Suppose we don't know the math to answer this question. We are told it's a rectangle, so we know the angles must be right angles, and we know the area must be length (AD) times height (AB). We know the height is 10. We know AD is drawn to scale. It definitely is longer than AB, so the area is definitely larger than 10 x 10 (answer (A) is out). AD doesn't look as long as twice AB, so the area is definitely less than 10 x 20 (answers (D) & (E) are out). Notice, with pure spatial estimation, we eliminated three of the five answer choices, so it will be to our advantage to guess randomly from the remaining two if we can't decide between them. Estimating from size can be a huge help if you don't remember the way to solve the problem.

BTW, the real math solution to that question: from the properties of the 30-60-90 triangle (ACD), we know that $$AD = 10*\sqrt{3}$$, and since $$\sqrt{3}$$ is approximately 1.7, AD is approximately 17, and the area is approximately 170.

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Re: The area of rectangle ABCD is closest to which of the following?  [#permalink]

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10 Jan 2018, 19:24
Bunuel wrote:

The area of rectangle ABCD is closest to which of the following?

(A) 100
(B) 130
(C) 170
(D) 200
(E) 230

Attachment:
drawn-1.png

Kudos for a correct solution.

--------------------
Let the length of the diagonal be h.
h*cos 60 = 10 (=b)
h*(1/2)=10
h=20

l*l + b*b =20*20
since b =10
=> l = 10 * sqrt(3)
Area =l*b = 10 * sqrt(3) * 10
=100 * sqrt(3)
=100 * 1.732
=173.2 closest to 170
Re: The area of rectangle ABCD is closest to which of the following?   [#permalink] 10 Jan 2018, 19:24
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