Trapped with this trapezium

Agnishom · 2013-01-19 16:46:49

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

How?

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

Bob · 2013-01-19 21:53:14

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 + 2ABCD

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

bobbym · 2013-01-19 23:25:18

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?

Bob · 2013-01-19 23:27:45

hi bobbym,

Wow! Are we in telepathic communication or what ?

My edit was at your time less 20 seconds.

Bob

bobbym · 2013-01-19 23:37:06

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.

Bob · 2013-01-19 23:44:41

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

bobbym · 2013-01-20 00:54:30

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

Agnishom · 2013-01-20 01:04:22

Let me check....

Agnishom · 2013-01-20 01:05:57

Sorry, it should have been AC² + BD² = BC² + AD² + 2ABCD

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)

bobbym · 2013-01-20 01:18:35

Hi Agnishom;

That one I believe works, yes post the hint.

Agnishom · 2013-01-20 01:25:33

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

Bob · 2013-01-20 03:40:45

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

Bob

scientia · 2013-01-20 08:25:15

Hence

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

Agnishom · 2013-01-20 14:47:34

Got it .... Thanks

Math Is Fun Forum

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

Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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

Re: Trapped with this trapezium

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