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

Agnishom
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From: Riemann Sphere
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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

bob bundy
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Registered: 2010-06-20
Posts: 7,193

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


Children are not defined by school ...........The Fonz
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: 97,276

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.

If it ain't broke, fix it until it is.

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

bob bundy
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Registered: 2010-06-20
Posts: 7,193

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


Children are not defined by school ...........The Fonz
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: 97,276

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.

If it ain't broke, fix it until it is.

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

bob bundy
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Registered: 2010-06-20
Posts: 7,193

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


Children are not defined by school ...........The Fonz
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
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 97,276

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.

If it ain't broke, fix it until it is.

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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 22,016
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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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 22,016
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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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 97,276

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.

If it ain't broke, fix it until it is.

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 22,016
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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,193

Re: Trapped with this trapezium

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

Bob


Children are not defined by school ...........The Fonz
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: 224

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: Riemann Sphere
Registered: 2011-01-29
Posts: 22,016
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'
'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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