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# a parallelogram has 2 sides: 14, 18 one of the diagonals is

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a parallelogram has 2 sides: 14, 18 one of the diagonals is [#permalink]  24 Sep 2005, 06:03
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a parallelogram has 2 sides: 14, 18 one of the diagonals is 16, what is the length of the othe diagonal?
Senior Manager
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Is there some data missing, like an angle? I'm not sure if there is sufficient information, unless, there is a property of a parallelogram that I missed.

How do we solve this or whats the OE?
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-Vikram

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Length of the second diagnol is 44.

Area(parallelogram)=18X14=252/2=176 (Area of each triangle)

Base/2Xheight=Area(triangles)

16/2XH=176

H=22

22X2=44
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GMATT73 wrote:
Length of the second diagnol is 44.

Area(parallelogram)=18X14=252/2=176 (Area of each triangle)

Base/2Xheight=Area(triangles)

16/2XH=176

H=22

22X2=44

Area of a parallelogram = base * height. How did you calculate height?
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anymore guys. the question reminds you an important concept/formulae.
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i give a shot : 2sqreroot33
but actually it's really because I want to participate
Come on, what's the trick ?
Manager
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It's a very good question.
lets a and b - sides
d1 and d2 diagonals
The key formular is:
d1^2+d2^2=2*(a^2+b^2).

from that another digonal is 28 ( if I'm not mistaken in calculations).
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I don't think there is sufficient info.
Himalaya, do u have an answer?
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Quote:
d1^2+d2^2=2*(a^2+b^2).

I'm not sure if that formula applies to a parallelogram. Looks more like what would apply to a kite (adjacent sides equal and diagonals perpendicular).

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If anyone doesn't believe the answer is 28,
you can work it out by drawing all the right angles in and applying Pythagoras' theorem several times and substituting.

So draw a rectangle which is 18+x wide, y high. In the top left take out a triangle x wide, with hypotenuse of 14. Then 14^2 = y^2 + x^2

and so on
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How do the diagonals come in the picture?
Richard.. can you elaborate on that? I tried like you said but somehow the diagonals dont get included on any of the pythagorean equations... What am I missing?

Thanks.
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-Vikram

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Difficult to describe.

Draw a big rectangle on a sheet of paper.
Its height is y.
Its length is 18+x.
On the top mark x from the left.
On the bottom mark x from the right.
Draw in the parallelogram inscribed within this rectangle.
(By drawing line from bottom left corner to point on top x from left.)
Mark a rectangle on the left, which is x wide, y high.
The diagonal is 14. Side of parallelogram.

So x^2 + y^2 = 14^2 = 196 (E1)

Let z = 18 - x (E2), z is the middle width portion of the rectangle.

Then you can split the bottom line, into three parts,
x, z, x.
Now we know the length of the shorter diagonal 16.
But this is the hypotenuse of a triangle.
The sides are y and z.

So y^2 + z^2 = 16^2 = 256 (E3)

(E3) - (E1)
z^2 - x^2 = 60
From (E2) z = 18 - x
So z^2 = 18^2 - 36x + x^2

(324 - 36x + x^2) - x^2 = 60
(324 - 60) = 36x
x = 264 / 36 = 22 / 3

Hence (E2)
z = 18 - 22/3 = 32/3

And (E1)
y^2 = 196 - x^2 = 196 - (22/3)^2
y^2 = (1764 - 484) / 9 = 1280/9

Width of rectangle,
w = 18 + x = (54+22)/3 = 76/3

Diagonal d
d^2 = w^2 + y^2
d^2 = (76^2)/9 + 1280/9
d^2 = (5776+1280)/9 = 7056/9
d = 84/3 = 28

But there would not be time to do this in the real GMAT.
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Richard,
That did clear my confusion. Thanks for the detailed explanation.
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vikramm wrote:
Quote:
d1^2+d2^2=2*(a^2+b^2).

I'm not sure if that formula applies to a parallelogram. Looks more like what would apply to a kite (adjacent sides equal and diagonals perpendicular).

I too was wondering where from this new formula came. A little trignometry helped derive formula though. Anyone interested in derivation can look into attached doc.

For others, get this formula in your kitty.
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