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The parallelogram shown has

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by boomgoesthegmat » Thu May 19, 2016 3:43 pm
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The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal?

A) 1/2

B) 1/root2

C) 1/2root2

D) 1/root3

E) 1/2root3

OA: D
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by OptimusPrep » Thu May 19, 2016 7:56 pm
boomgoesthegmat wrote:Image

The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal?

A) 1/2

B) 1/root2

C) 1/2root2

D) 1/root3

E) 1/2root3

OA: D
Assume the length of each side = x

The other angles would be 60, 120 and 120.
Diagonal divides the parallelogram into half

Hence the shorter diagonal will make two equilateral triangles
Length of the shorter diagonal = length of side = x

For the longer diagonal.
The diagonal will divide the parallelogram in to isosceles triangles with angles 30, 120, 30
Dropping a perpendicular from top most point on to the diagonal, we have two triangles with angles 30, 60 and 90 with base as half the length of diagonal

Hypotenuse = x,
Cos 30 = base/x
base = (√3/2)*x

Hence the length of the diagonal = √3x

Ratio of the shorter to the longer diagonal = x : √3x = 1: √3

Correct Option: D
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by Scott@TargetTestPrep » Mon Nov 20, 2017 4:11 pm
boomgoesthegmat wrote:Image

The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal?

A) 1/2

B) 1/root2

C) 1/2root2

D) 1/root3

E) 1/2root3
We are given a parallelogram with equal sides, and we must determine the ratio of the length of the shorter diagonal to that of the longer diagonal. Since the sides are all equal, we know we have a rhombus, and the diagonals are perpendicular. Let's sketch this diagram below.

Image

We should see that the diagonals bisect each angle of the rhombus, and thus we have created four 30-60-90 right triangles. Using our side ratio of a 30-60-90 right triangle, we have:

x : x√3 : 2x

Let's use this side ratio to determine the lengths of each diagonal in terms of x.

Image

We can see that the length of the shorter diagonal is x + x = 2x, and the length of the longer diagonal is x√3 + x√3 = 2x√3. Thus, the ratio of the length of the shorter diagonal to the length of the longer diagonal is:

(2x)/(2x√3) = 1/√3

Answer: D

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