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April 17th, 2014, 11:35 PM   #1
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Exclamation Why area differs if approach is changed ?

Hi friends,

I am sorry if this not the right section to ask this. I was solving a simple calculus math and was about to get the answer, but got interrupted by my study-mate (is it a word ?) and then confusion came like :-

Figure is like this :



A quadrilateral ABCD where AB || CD and
AB = 78 CM
CD = 52 CM
AD = 28 CM
BC = 30 CM

We need to find the area of this quadrilateral.

My approach :
Drawing a line CE so, it creates a parallelogram with AECD and
AE = 52
CE = 28
CD = 52
AD = 28

From, here I went to the Triangle BCE where
BC = 30
CE = 28 (from above)
EB = 26 (78-52)

Then, I calculated the area of this Triangle using S and tried to find the h of this triangle which came like :

336/13 CM

Now, as I have the distance or height of the quadrilateral ABCD as 336/13
I calculated the area using formula 1/2 (AB+CD) * d (or h)

Which is 1680 CM^2, and correct by book itself.

But my friend interrupted when I calculated the area of The Triangle BCE and said that as we have the area of Triangle, we have to only calculate the area of the parallelogram and adding both we will get the required area, it sound right to me, so I tried again with his approach where now I had a parallelogram like :

AECD, where
AE = 52
CE = 28
CD = 52
AD = 28

using formula H*W = 52*28 = 1456 CM^2

And, we had already Triangles area as 336 CM^2

And theoretically the area of the quadrilateral should be equal to = 1456+336
=1752 CM^2

which is not correct as I calculated above, and book also confirmed 1680 CM^2.

Any idea why both differs so much (I could understand if it has minor degree varying).
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April 18th, 2014, 01:24 AM   #2
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You used the wrong H for the parallelogram, you should have used h (336/13)
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April 18th, 2014, 04:53 AM   #3
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Quote:
Originally Posted by weirddave View Post
You used the wrong H for the parallelogram, you should have used h (336/13)
Ok my bad, but its same H or h, just I wrote H in hurry.
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April 19th, 2014, 07:38 AM   #4
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*Bump*
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April 19th, 2014, 08:09 AM   #5
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Your friend is wrong. The area of the parallelogram is not 52*28, but in general

$\displaystyle area = ab\sin (\textrm{angle between sides})$.

Only, if the angle is $\displaystyle 90^{\circ}$, the area is $\displaystyle ab = ah$.
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April 19th, 2014, 08:14 AM   #6
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Quote:
Originally Posted by Amanalice View Post
Then, I calculated the area of this Triangle using S and tried to find the h of this triangle which came like :

336/13 CM
You made a mistake here: $h$ should be $\mathrm{\dfrac{198}{13}\ cm}$. $\mathrm{\dfrac{336}{13}\ cm}$ is the length of EB.
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April 19th, 2014, 09:19 AM   #7
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Shit, Now I got it too. Dont know how it got slipped from my logic.

Very much thank you friends.

Feeling happy
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April 19th, 2014, 01:36 PM   #8
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Quote:
Originally Posted by Amanalice View Post
using formula H*W = 52*28 = 1456 CM^2

And, we had already Triangles area as 336 CM^2

And theoretically the area of the quadrilateral should be equal to = 1456+336
=1752 CM^2

The H used was 28, whereas it should have been 336/13 (the same h as the triangle), sorry if it wasn't clear that's what I meant in my previous post.
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