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# Each edge of the cube shown above has length s. What is the perimeter

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Joined: 02 Sep 2009
Posts: 43363
Each edge of the cube shown above has length s. What is the perimeter [#permalink]

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30 Nov 2017, 21:52
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Each edge of the cube shown above has length s. What is the perimeter of ∆ BDE?

(A) 3s
(B) 6s
(C) s√3/2
(D) 3s√2
(E) 2s + s√2

[Reveal] Spoiler:
Attachment:

2017-12-01_0947.png [ 9.8 KiB | Viewed 378 times ]
[Reveal] Spoiler: OA

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Re: Each edge of the cube shown above has length s. What is the perimeter [#permalink]

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30 Nov 2017, 21:56
D
By Pythagoras theorem we have BD=BE=DE=s*(2)^0.5. Hence the perimeter of triangle BDE is 3*s*(2)^0.5

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Re: Each edge of the cube shown above has length s. What is the perimeter [#permalink]

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30 Nov 2017, 22:09
Bunuel wrote:

Each edge of the cube shown above has length s. What is the perimeter of ∆ BDE?

(A) 3s
(B) 6s
(C) s√3/2
(D) 3s√2
(E) 2s + s√2

[Reveal] Spoiler:
Attachment:
2017-12-01_0947.png

In a cube all the sides are equal. Consider ABCD as square

Apply Pythagoras theorem BE=ED= BD = $$\sqrt{2}$$s

Hence Perimeter =3s$$\sqrt{s}$$

Hence D
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Re: Each edge of the cube shown above has length s. What is the perimeter [#permalink]

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01 Dec 2017, 12:39
Bunuel wrote:

Each edge of the cube shown above has length s. What is the perimeter of ∆ BDE?

(A) 3s
(B) 6s
(C) s√3/2
(D) 3s√2
(E) 2s + s√2

[Reveal] Spoiler:
Attachment:
2017-12-01_0947.png

Each side of ∆ BDE is the hypotenuse of a two-dimensional isosceles right triangle.
They are congruent, and there are three: ∆ BCD, ∆ CDE, and ∆ BCE

Each has side length $$s$$ (= BC, CD, and CE)

Because they are one-half of a square:
All have angle measures of 45-45-90 and
corresponding side lengths*
$$x : x : x\sqrt{2}$$

Side/leg length $$x = s$$
Hypotenuse, per ratio, hence is $$s\sqrt{2}$$ , which =
Length of all three sides of ∆ BDE

Perimeter of ∆ BDE
Three sides of length $$s\sqrt{2}$$ =
$$3s\sqrt{2}$$

*Knowing those ratios is key, but if not:
A square cut by a diagonal produces two right isosceles triangles
The relationship between one leg of such a triangle its hypotenuse (also the square's diagonal)

$$h = s\sqrt{2}$$ , derived from Pythagorean theorem:
$$s^2 + s^2 = h^2$$
$$2s^2 = h^2$$
$$\sqrt{2}\sqrt{s^2} = \sqrt{h^2}$$
$$\sqrt{2}s = h$$
$$h = s\sqrt{2}$$

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Re: Each edge of the cube shown above has length s. What is the perimeter   [#permalink] 01 Dec 2017, 12:39
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# Each edge of the cube shown above has length s. What is the perimeter

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