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# The above cube has sides of length 8*2^(1/2). What is the perimeter of

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Joined: 02 Sep 2009
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The above cube has sides of length 8*2^(1/2). What is the perimeter of  [#permalink]

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04 Sep 2018, 23:19
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Difficulty:

25% (medium)

Question Stats:

77% (00:51) correct 23% (01:27) wrong based on 28 sessions

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

A. 24

B. $$24\sqrt{2}$$

C. 48

D. $$48\sqrt{2}$$

E. $$48\sqrt{3}$$

Attachment:

image001.gif [ 705 Bytes | Viewed 354 times ]

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Re: The above cube has sides of length 8*2^(1/2). What is the perimeter of  [#permalink]

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05 Sep 2018, 00:34
Bunuel wrote:

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

A. 24

B. $$24\sqrt{2}$$

C. 48

D. $$48\sqrt{2}$$

E. $$48\sqrt{3}$$

Attachment:
image001.gif

Perimeter of triangle PQR=PQ+QR+RP, where PQ=QR=RP are the diagonals of the faces of the cube.

Diagonal of any face of cube=$$\sqrt{2}$$* side length=$$\sqrt{2}*8\sqrt{2}$$=16

So, perimeter=16+16+16=48

Ans. (C)
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The above cube has sides of length 8*2^(1/2). What is the perimeter of  [#permalink]

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05 Sep 2018, 16:27
Bunuel wrote:

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

A. 24

B. $$24\sqrt{2}$$

C. 48

D. $$48\sqrt{2}$$

E. $$48\sqrt{3}$$

Attachment:
image001.gif

The perimeter of triangle PQR is the sum of the lengths of three diagonals of identical square faces.

The diagonal of a square cuts it into two congruent 45-45-90 triangles with side lengths opposite those angles in the ratio $$x : x : x \sqrt{2}$$

Side length of $$8\sqrt{2}$$ corresponds with $$x$$.

The diagonal, $$d$$, thus corresponds with $$x\sqrt{2}$$
$$d= (8\sqrt{2}*\sqrt{2})=(8*2)=16$$

Or use Pythagorean theorem, $$s$$ = side
$$s^2+s^2=d^2$$
$$(8\sqrt{2}*8\sqrt{2})+(8\sqrt{2}*8\sqrt{2})= d^2$$
$$(128+128)=d^2$$
$$\sqrt{d^2}=\sqrt{256}=\sqrt{2^8}=\sqrt{(2^4)^2}=2^4$$
$$d=16$$

Perimeter of triangle PQR:
$$(16+16+16)=48$$

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The above cube has sides of length 8*2^(1/2). What is the perimeter of   [#permalink] 05 Sep 2018, 16:27
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