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#1 2013-01-19 16:46:49

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
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Trapped with this trapezium

In a trapezium ABCD, if AB||CD then AC² + BD² = BC² + AB² + 2•AB•CD

How?

Last edited by Agnishom (2013-01-19 16:48:11)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#2 2013-01-19 21:53:14

bob bundy
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Registered: 2010-06-20
Posts: 6,471

Re: Trapped with this trapezium

hi Agnishom,

I haven't fully explored this yet; maybe later;  but it has the look of several applications of the cosine rule.  As no cos(X) terms exist they must be cancelling out.  Remember that angle ABD = angle BDC because of the parallels.

http://www.mathsisfun.com/algebra/trig-cosine-law.html

So perhaps you could have a go yourself ?  smile

LATER EDIT:

I've tried what I suggested and reached a formula connecting AC, BD, BC, AB, CD.  But it isn't this one:

AC² + BD² = BC² + AB² + 2•AB•CD

So I set up the diagram on Sketchpad and got (for my trapezium)

LHS = 162.86  and RHS = 193.64

Are you sure you've got the right identity ?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2013-01-19 23:25:18

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 89,150

Re: Trapped with this trapezium

Hi Agnishom;

How?

You might have a hard time proving that because I have been playing with trapeziums in geogebra and I can not find a single one where that is true.

Did you copy the problem correctly? Or what am I missing?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#4 2013-01-19 23:27:45

bob bundy
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Registered: 2010-06-20
Posts: 6,471

Re: Trapped with this trapezium

hi bobbym,

Wow!  Are we in telepathic communication or what ?

My edit was at your time less 20 seconds.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2013-01-19 23:37:06

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 89,150

Re: Trapped with this trapezium

Hi Bob;

Looks that way!

I have been looking at it for a long time, I wanted to be sure. I thought maybe I was doing something wrong.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#6 2013-01-19 23:44:41

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,471

Re: Trapped with this trapezium

hi bobbym

I thought maybe I was doing something wrong.

That's the difference between us then.  I tend to blunder in and post anyway.  I guess that's why my posts are full of corrections.

Four applications of the cosine rule does allow you to eliminate any angles and get an equation.  It isn't that one; not even close.

That's as far as I've got.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#7 2013-01-20 00:54:30

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 89,150

Re: Trapped with this trapezium

Not so different. I blunder around plenty but Agnishom is usually careful.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#8 2013-01-20 01:04:22

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,222
Website

Re: Trapped with this trapezium

sad Let me check....


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#9 2013-01-20 01:05:57

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,222
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Re: Trapped with this trapezium

Sorry, it should have been AC² + BD² = BC² + AD² + 2•AB•CD

P.S.: I have a hint in my book. But I dont get it. Do you want it?

Last edited by Agnishom (2013-01-20 01:08:14)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#10 2013-01-20 01:18:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,150

Re: Trapped with this trapezium

Hi Agnishom;

That one I believe works, yes post the hint.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#11 2013-01-20 01:25:33

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,222
Website

Re: Trapped with this trapezium

Let CD < AB,
Drop perpendiculars DE and CF from D and C respectively.

In ΔABC, angle B is acute
Therefore, AC^2 = BC^2 + AB^2 - 2*AB*AE
In ΔABD, angle A is acute
Therefore, BD^2 = AD^2 + AB^2 - 2*AB*AF


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#12 2013-01-20 03:40:45

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,471

Re: Trapped with this trapezium

Now my diagram isn't making sense.  Is the lettering A to B to C to D ?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#13 2013-01-20 08:25:15

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: Trapped with this trapezium

Trapezium.gif

Hence

Last edited by scientia (2013-01-20 08:26:55)

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#14 2013-01-20 14:47:34

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 18,222
Website

Re: Trapped with this trapezium

Got it .... Thanks smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

Offline

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