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The parallelogram shown has four sides of equal length. What is the

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New post 21 Oct 2015, 00:19
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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) \(\frac{1}{\sqrt{2}}\)

(C) \(\frac{1}{2\sqrt{2}}\)

(D) \(\frac{1}{\sqrt{3}}\)

(E) \(\frac{1}{2\sqrt{3}}\)

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2015-10-21_1114.png
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 20 Nov 2017, 12:38
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Bunuel 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) \(\frac{1}{\sqrt{2}}\)

(C) \(\frac{1}{2\sqrt{2}}\)

(D) \(\frac{1}{\sqrt{3}}\)

(E) \(\frac{1}{2\sqrt{3}}\)

Kudos for a correct solution.

Attachment:
2015-10-21_1114.png


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

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New post 08 Jun 2016, 04:31
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The parallelogram shown has four sides of equal length.

Inference : It is a rhombus.

Properties of Rhombus :

1) Diagonals are not congruent

2) Diagonals act as angle bisectors

3) Diagonals intersect at right angles and bisect each other

Hence we get a 30-60-90 triangle as shown below. Characteristic property of this triangle is that sides are in the ratio of 1:[ sqrt 3 ]:2

Shortest diagonal is the diagonal opposite the 30 degree angle and longest diagonal is the the one opposite 60 degree angle

Therefore ratio of shortest to longest is 1: sqrt 3
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The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 21 Oct 2015, 01:15
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IMO D.

The parallelogram has four equal sides (x), if the angles is 60° for one, the opposite angle is 60° and the other two are 120° each (360-60-60/2).

Therefore, the parallelogram diagonal bisects every angle to form four different 30-60-90 rectangle triangles (ratio: \(x : 2x : x\sqrt{3}\)).

Longer diagonal = \(2x\sqrt{3}\)
Shorter diagonal = \(2x\)

Therefore the ratio is \(\frac{2x}{2x\sqrt{3}}\) = \(\frac{1}{\sqrt{3}}\)
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New post 21 Oct 2015, 01:29
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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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New post 21 Oct 2015, 05:00
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).

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

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New post 21 Oct 2015, 09:17
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Bunuel wrote:
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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) \(\frac{1}{\sqrt{2}}\)

(C) \(\frac{1}{2\sqrt{2}}\)

(D) \(\frac{1}{\sqrt{3}}\)

(E) \(\frac{1}{2\sqrt{3}}\)

Kudos for a correct solution.

Attachment:
2015-10-21_1114.png


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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New post 03 Apr 2016, 21:28
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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New post 06 Jun 2016, 08:44
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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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 08 Jun 2016, 04:53
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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

so the ratio= 1/ (3)^0.5
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 05 Oct 2016, 01:05
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)

Is that a math rule i've forgotten?

Please help thank you !
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 05 Oct 2016, 02:09
xnthic wrote:
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)

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.

For more check here: math-polygons-87336.html

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

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New post 25 Mar 2017, 10:17
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
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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Attached is a visual that should help.
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 30 May 2017, 22:44
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Bunuel wrote:
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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) \(\frac{1}{\sqrt{2}}\)

(C) \(\frac{1}{2\sqrt{2}}\)

(D) \(\frac{1}{\sqrt{3}}\)

(E) \(\frac{1}{2\sqrt{3}}\)

Kudos for a correct solution.

Attachment:
2015-10-21_1114.png


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

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New post 16 Nov 2017, 19:57
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A low level approach of assuming that each side is of length 1 and that we have two equilateral triangles joint together:
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 23 Aug 2018, 05:08
I get 1/root(2). Can somebody please help me to explain where I am wrong.

Here is my approach for the long diagonal. So I know that I have a triangle with the following facts: 1 Angle 120° and two times 30°. Two sides have length x therefore my third side (the long diagonal) must be x*root(2).
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Re: The parallelogram shown has four sides of equal length. What is the  [#permalink]

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New post 23 Aug 2018, 08:38
HHPreparation wrote:
I get 1/root(2). Can somebody please help me to explain where I am wrong.

Here is my approach for the long diagonal. So I know that I have a triangle with the following facts: 1 Angle 120° and two times 30°. Two sides have length x therefore my third side (the long diagonal) must be x*root(2).


The sides are in the ratio \(1:1:\sqrt{2}\) only when the angles are 45-45-90. It is not the same when the angles are 30-30-120. There is no "hypotenuse" here.
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