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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

### 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'
I'm not crazy, my mother had me tested.

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,972

### 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 ?

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
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
Always satisfy the Prime Directive of getting the right answer above all else.

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,972

### 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
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
Always satisfy the Prime Directive of getting the right answer above all else.

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,972

### 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
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
Always satisfy the Prime Directive of getting the right answer above all else.

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

### Re: Trapped with this trapezium

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'
I'm not crazy, my mother had me tested.

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
Website

### 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'
I'm not crazy, my mother had me tested.

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
Always satisfy the Prime Directive of getting the right answer above all else.

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

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,773
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'
I'm not crazy, my mother had me tested.

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

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,972

### 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

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: 24,773
Website

### Re: Trapped with this trapezium

Got it .... Thanks

'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'
I'm not crazy, my mother had me tested.

Offline