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# Equilateral triangle AED lies inside square ABCD, as shown above. If

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Math Expert
Joined: 02 Sep 2009
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Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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21 Oct 2016, 00:15
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Difficulty:

25% (medium)

Question Stats:

81% (01:08) correct 19% (01:14) wrong based on 83 sessions

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Equilateral triangle AED lies inside square ABCD, as shown above. If the side of the square is 6, what is the area of the shaded region?

A. $$19-9\sqrt{3}$$

B. $$36-18\sqrt{3}$$

C. $$9\sqrt{3}$$

D. $$3(12-3\sqrt{3})$$

E. $$2(9+3\sqrt{3})$$

Attachment:

T7902.png [ 7.59 KiB | Viewed 2185 times ]

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Re: Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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21 Oct 2016, 13:02
1
Bunuel wrote:

Equilateral triangle AED lies inside square ABCD, as shown above. If the side of the square is 6, what is the area of the shaded region?

A. $$19-9\sqrt{3}$$

B. $$36-18\sqrt{3}$$

C. $$9\sqrt{3}$$

D. $$3(12-3\sqrt{3})$$

E. $$2(9+3\sqrt{3})$$

Attachment:
T7902.png

Area of shaded region = Area of Square - Area of the Equilateral Triangle

Area of the square $$= 6^2 = 36$$

Area of the Equilateral Triangle =$$\frac{√3}{4}*6^2$$ = $$9√3$$

So, Area of shaded region = $$36$$ $$-$$ $$9√3$$

Or, Area of shaded region = $$3(12-3\sqrt{3})$$

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Re: Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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21 Oct 2016, 13:36
Bunuel wrote:

Equilateral triangle AED lies inside square ABCD, as shown above. If the side of the square is 6, what is the area of the shaded region?

A. $$19-9\sqrt{3}$$

B. $$36-18\sqrt{3}$$

C. $$9\sqrt{3}$$

D. $$3(12-3\sqrt{3})$$

E. $$2(9+3\sqrt{3})$$

Attachment:
T7902.png

the side of the equilateral triangle is 6 as theside of the square.

the height of an equilateral triangle is equal to $$\frac{side}{2}*\sqrt{3}$$ = $$3\sqrt{3}$$

the area of the triangle is $$\frac{1}{2}*6*3*\sqrt{3}$$ = $$9\sqrt{3}$$

the area of the square is $$6*6 = 36$$

the difference between the area of the square and the area of the triangle is $$36-$$$$9\sqrt{3}$$ $$= 3*(12+3\sqrt{3})$$

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Re: Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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21 Oct 2016, 18:48
Bunuel wrote:

Equilateral triangle AED lies inside square ABCD, as shown above. If the side of the square is 6, what is the area of the shaded region?

A. $$19-9\sqrt{3}$$

B. $$36-18\sqrt{3}$$

C. $$9\sqrt{3}$$

D. $$3(12-3\sqrt{3})$$

E. $$2(9+3\sqrt{3})$$

Attachment:
T7902.png

Area of the square = 6*6 = 36
Since the triangle is an equilateral triangle and shares one side with the square, side of the triangle = 6

Area of the triangle = (√3/4)*6^2 = 9√3

Area of the shaded region = 36 - 9√3 = 3(12 - 3√3)

Correct Option: D
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Posts: 80
Re: Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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22 Oct 2016, 04:55
This seems to be formula based

Area of square = a ^2 = 36

Area of equilateral triangle = Sq rt 3/4 * a^2
= 9 * sq rt 3

Area of shaded region = 36 - 9*sq rt 3
= 3(12−3√3)

Option D

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Re: Equilateral triangle AED lies inside square ABCD, as shown above. If  [#permalink]

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08 Jul 2018, 23:12
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Re: Equilateral triangle AED lies inside square ABCD, as shown above. If &nbs [#permalink] 08 Jul 2018, 23:12
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