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

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

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New post 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
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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New post 30 Nov 2017, 21:56
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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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New post 30 Nov 2017, 22:09
Bunuel wrote:
Image
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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New post 01 Dec 2017, 12:39
Bunuel wrote:
Image
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 \(s\sqrt{2}\) =
\(3s\sqrt{2}\)

Answer D

*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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