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# If the diagonal of a square of side √6 is the side of an equilateral

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Math Expert
Joined: 02 Sep 2009
Posts: 43808
If the diagonal of a square of side √6 is the side of an equilateral [#permalink]

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01 Aug 2017, 11:56
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73% (00:57) correct 27% (01:12) wrong based on 51 sessions

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If the diagonal of a square of side √6 is the side of an equilateral triangle, what is the area of this triangle?

A. (3/8)·√3
B. (3/4)·√3
C. (3/2)·√3
D. 2·√6
E. 3·√3

[Reveal] Spoiler:
Attachment:

T6030.png [ 3.75 KiB | Viewed 1209 times ]
[Reveal] Spoiler: OA

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If the diagonal of a square of side √6 is the side of an equilateral [#permalink]

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05 Aug 2017, 17:26
Bunuel wrote:

If the diagonal of a square of side √6 is the side of an equilateral triangle, what is the area of this triangle?

A. (3/8)·√3
B. (3/4)·√3
C. (3/2)·√3
D. 2·√6
E. 3·√3

[Reveal] Spoiler:
Attachment:
T6030.png

The diagonal of a square is s√2, because the diagonal is the hypotenuse of a 45-45-90 triangle, with side ratio $$x: x : x\sqrt{2}$$

If square side is √6, its diagonal is √6 * √2 = √12

Side of equilateral triangle therefore is √12. Formula for area of equilateral triangle is

$$\frac{s^2\sqrt{3}}{4}$$

$$\frac{(√12)^2 * \sqrt{3}}{4}$$ =

$$\frac{12\sqrt{3}}{4}$$ =

3√3

If you don't know or don't remember the formula for the area of an equilateral triangle, draw its median (straight line from one vertex to opposite segment). The median of an equilateral triangle is a perpendicular bisector (of both angle and opposite side), and the altitude. You'll have two 30-60-90 triangles. Find the area of the equilateral triangle from there.
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Formerly genxer123

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Joined: 07 Jun 2017
Posts: 108
Re: If the diagonal of a square of side √6 is the side of an equilateral [#permalink]

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05 Aug 2017, 23:41
the height of the equilateral triangle is side* (sqrt 3/2)
so [sqrt 6*(sqrt 3/2) * sqrt 6 ]/2
so it is 3 sqrt 3
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Joined: 25 Jul 2017
Posts: 7
Re: If the diagonal of a square of side √6 is the side of an equilateral [#permalink]

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06 Aug 2017, 00:49
Diagonal = side of equilateral triangle.
Hence, diagonal = 2sqrt(3) by Pythagoras theorem on square of side sqrt(6)

Now, aread of triangle = sqrt(3)/4 (side)^2 = 3sqrt(3).
Hence, option E
Re: If the diagonal of a square of side √6 is the side of an equilateral   [#permalink] 06 Aug 2017, 00:49
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