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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The parallelogram shown has
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boomgoesthegmat
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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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OptimusPrep
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Assume the length of each side = xboomgoesthegmat wrote:
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
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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Scott@TargetTestPrep
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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.boomgoesthegmat wrote:
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 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.
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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