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The parallelogram shown has four sides of equal length. What is the [#permalink]

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21 Oct 2015, 00:29

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Opposite angles of a parallegram are equal and sum of adjacent angles is 180 degree . Each of the angles adjacent to 60 is 120 . The shorter diagonal divies the parallelogram divides into 2 isosceles triangles . Since all the sides of the parallelogram are equal , we get an equilateral triangle. Therefore the shorter diagonal will be equal to length of sides of parallelogram = x

The longer diagonal divides parallelogram into 2 isoceles trianges with one angle 120 and the other 2 angles equal to 30 .

Since, the diagonals of a parallelogram are perpendicular bisectors ,the 2 diagonals divide the parallelogram into 4 - 30-60-90 triangles.

The sides of a 30-60- 90 triangle are in ratio 1:(3^(1/2)):2 Sin 60=(y/2)/x Where y= length of the longer diagonal =>(3^(1/2))/2=(y/2)/x =>x/y=1/(3^(1/2))

Answer D
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Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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21 Oct 2015, 04:00

Let ABCD be our parallelogram with A = 60 degree and AB=BC=CD=AD=1 Opposite angles of a parallelogram are equal. So C = 60 degree. Sum of adjacent angles = 180 degree. So B = 120 degree and D = 120 degree. AC and BD be the diagonal. Applying cosine rule (a^2 = b^2 + c^2 - 2bc(cos(included angle)), AC^2 = 1 + 1 - 2cos(120) = 2 - 2(-0.5) = 3 --> AC = sqrt(3) BD^2 = 1 + 1 - 2cos(60) = 2 - 2(0.5) = 1 --> BD = 1

Length of shorter diagonal to length of longer diagonal = 1:sqrt(3).

The figure must be a rhombus. Draw the diagonals on your scratch paper. You will now have two equiliteral triangles. The shorter diagonal is one side of such an equiliteral triangle. Let's say s=1.

To get the longer side of the diagonal of the rhombus, you have 2*height of the equiliteral triangle. If you split an equiliteral triangle into two parts, you will have two 30:60:90 triangles. The height of the equiliteral triangle if s = 1 is \(0.5*\sqrt{3}\) and the long diagonal will be \(2*0.5*\sqrt{3}\)

So you have \(1:\sqrt{3}\)

Answer D
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Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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03 Apr 2016, 20:28

Here is what i did Firstly the given ||gm has all sides equal => Rhombus

Now shorter diagonal is equal to the side of the ||gm as the equilateral triangle is formed. Now to find the Longer diagonal i used the property that diagonals of rhombus bisect each other at a right angle.. so longer diagonal => √3 x side hence the ratio => 1/3 or 1:3
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Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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06 Jun 2016, 07:44

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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

Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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08 Jun 2016, 03:53

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Lets assume that side = 1 now the shorter diagonal will belong to squeezed quadrilateral with one side = 1 - 1/2 = 1/2( being a 30-60-90 triangle on the right side) & one side = (3)^0.5/2 Longer diagonal will belong to enlarged quadrilateral with one side = 1 + 1/2 = 3/2 (being a 30-60-90 triangle but side will add) & one side = (3)^0.5/2

to find the diagonal we will apply Pythagoras to both rectangles Squeezed Quadrilateral : ( 1/4 + 3/4) ^ 0.5 = 1 Enlarged Quadrilateral : ( 9/4 + 3/4 ) ^ 0.5 = 3^0.5

Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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05 Oct 2016, 00:05

A novice question:

For parallelogram, i understand the rules of relationship for 30:60:90, opposite angels are congruent etc.

But how can we ascertain that when we split the parallelogram in to two parts , the 60 and 120 degrees angles on the vertices split in to exactly HALF? (i.e. 30 , 60 degrees)

For parallelogram, i understand the rules of relationship for 30:60:90, opposite angels are congruent etc.

But how can we ascertain that when we split the parallelogram in to two parts , the 60 and 120 degrees angles on the vertices split in to exactly HALF? (i.e. 30 , 60 degrees)

Is that a math rule i've forgotten?

Please help thank you !

Usually diagonals of a parallelogram do not bisect the angles but here we have special kind of parallelogram - a rhombus, where the diagonals bisect the angles.

Re: The parallelogram shown has four sides of equal length. What is the [#permalink]

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25 Mar 2017, 09:17

the shorter diagonal is 1, because 60 can be calculated for the triangle->all sides are the same the longer one is square root of 3, because if we split triangles we will get to ration x:2x:x*(3)^1/2, because this is triangle of 30:60:90 angles 2x is 1. hence our side is 2*1/2*3^1/2 Answer is E

In a parallelogram, the opposite angles are equal. So angle opposite to 60 degrees is also 60. Draw the shorter diagonal. Since the sides are equal, we see that we get two congruent equilateral triangles. The shorter diagonal is the side of each equilateral triangle and the longer diagonal is twice the altitude of each equilateral triangle. We know that if the side of an equilateral triangle is s, its altitude is \(\sqrt{3}s/2\) and hence twice of its altitude is \(\sqrt{3}s\).

Required Ratio \(= s/\sqrt{3}s = 1/\sqrt{3}\)
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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.

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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