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The above cube has sides of length . What is the area of the triangle

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Bunuel
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The above cube has sides of length . What is the area of the triangle [#permalink] New post   12 Sep 2018, 00:27
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The above cube has sides of length \(8\sqrt{2}\). What is the area of the triangle PQR?


A. \(32\sqrt{3}\)

B. \(64\sqrt{2}\)

C. \(64\sqrt{3}\)

D. \(128\sqrt{2}\)

E. \(128\sqrt{3}\)


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Re: The above cube has sides of length . What is the area of the triangle [#permalink] New post   12 Sep 2018, 03:26
Bunuel wrote:

The above cube has sides of length \(8\sqrt{2}\). What is the area of the triangle PQR?


A. \(32\sqrt{3}\)

B. \(64\sqrt{2}\)

C. \(64\sqrt{3}\)

D. \(128\sqrt{2}\)

E. \(128\sqrt{3}\)


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image001.gif


PQ=QR=RP=diagonal of the faces(square shaped by birth) of the cube=√2*(8√2)=16

Now, triangle PQR is an equilateral triangle with sides PQ,QR,and RP.

Area of triangle PQR=√3/4*\(PQ^2\)=√3/4*16*16=64√3

Ans. (C)
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Re: The above cube has sides of length . What is the area of the triangle [#permalink] New post   12 Sep 2018, 05:30
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Bunuel wrote:

The above cube has sides of length \(8\sqrt{2}\). What is the area of the triangle PQR?


A. \(32\sqrt{3}\)

B. \(64\sqrt{2}\)

C. \(64\sqrt{3}\)

D. \(128\sqrt{2}\)

E. \(128\sqrt{3}\)


Show Spoiler
Attachment:
image001.gif


Point to Note: The triangle PQR is made of the diagonals of faces of cube i.e. the triangle PQR is an equilateral triangle with dimension a√2 where a is the side of cube

\(a = 8√2\)
i.e. Side of equilateral triangle \(= a√2 = 8√2*√2 = 16\)

Area of equilateral triangle pQR \(= (√3/4)Side^2 = (√3/4)16^2 = 64√3\)

Answer: option C
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The above cube has sides of length . What is the area of the triangle [#permalink] New post   12 Sep 2018, 05:58
The above cube has sides of length \(8\sqrt{2}\). What is the area of the triangle PQR?


A. \(32\sqrt{3}\)

B. \(64\sqrt{2}\)

C. \(64\sqrt{3}\)

D. \(128\sqrt{2}\)

E. \(128\sqrt{3}\)


Show Spoiler
Attachment:
image001.gif


All the sides of the triangle are the diagonal of faces of the cube. Hence, Sides of the triangle are equal and are equal to diagonal of the face of the cube.

Therefore, For the equilateral triangle PQR, The area ==√3/4*side2

the side of the triangle= diagonal of the cube = a√2, Plugging value, we get, diagonal= side of the triangle PQR=8√2∗√2=16

Area of triangle PQR =(√3/4)Side2

(√3/4)16^2=64√3

Hence C is the correct answer
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Re: The above cube has sides of length . What is the area of the triangle [#permalink] New post   14 Sep 2018, 09:05
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Bunuel wrote:

The above cube has sides of length \(8\sqrt{2}\). What is the area of the triangle PQR?


A. \(32\sqrt{3}\)

B. \(64\sqrt{2}\)

C. \(64\sqrt{3}\)

D. \(128\sqrt{2}\)

E. \(128\sqrt{3}\)


Show Spoiler
Attachment:
image001.gif


Since the side length of each square face of the cube is 8√2, then the length
of each diagonal is 8√2 x √2 = 16.

The 3 diagonals create equilateral triangle PQR, with base 16 and height
8√3. Thus, the area is:

(1/2) x 16 x 8√3 = 64√3

Answer: C
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Re: The above cube has sides of length . What is the area of the triangle [#permalink]
14 Sep 2018, 09:05
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