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

Question

https://gmatclub.com/forum/download/file.php?id=38963 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 2017-12-01_0947.png

Sasindran · Answer

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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NandishSS · Answer

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

generis · Answer

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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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

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