Proving Area Equality of Parallelograms ABCD & AXYZ

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Discussion Overview

The discussion revolves around proving the area equality of two parallelograms, ABCD and AXYZ, under specific conditions regarding the placement of points X and D. The focus is on vector representations and cross products to establish the relationship between the areas of the two shapes in a two-dimensional plane.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant outlines the problem of proving that the areas of parallelograms ABCD and AXYZ are equal given the positions of points X and D.
  • Another participant suggests using vector representation and the cross product to approach the proof.
  • A participant expresses difficulty in expressing vectors AX and AZ and encounters unknowns when attempting to compute the cross product.
  • There is a discussion on simplifying the expression for the cross product of AX and AZ and the goal of showing it equals the cross product of AB and AD.
  • One participant provides a series of vector equations but becomes stuck on proving the equality of certain vector expressions.
  • A geometric observation is proposed, suggesting a method of visualizing the areas through sliding and extending lines to demonstrate area equivalence.
  • Another participant encourages further exploration of the cross product definitions to aid in the proof process.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the proof method, and multiple approaches are being discussed without resolution. There is uncertainty regarding the simplification of vector expressions and the application of geometric observations.

Contextual Notes

Participants express challenges related to the representation of vectors and the complexity of the cross product calculations. The discussion includes unresolved mathematical steps and assumptions about the geometric configuration.

galois427
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the prob ask me to prove that if parallelograms ABCD and AXYZ (see the attachment) are such that point X lies on side BC and point D lies on side YZ, the area of the 2 parallelograms are equal.
 

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use vectors ...
AX = AB+BX
AZ = AD+DZ
take cross product of AX and AZ and simplify ...

-- AI
 
how would you express AX and AZ as a vector? I just set z=0 but when I took the cross product, i have a bunch of unknowns (x1,x2,x3,y0,y1).
 
we are working in the 2d plane so all vectors will be of form xi+yj tho we don't need to use this form here.
AX x AZ = (AB+BX) x (AD+DZ)
simplify this!
your goal is to show AX x AZ = AB x AD
tis a bit tricky but u can do it for sure!

-- AI
 
ok, this is what i got.

AX x AZ = (AB+BX) x (AD+DZ)
= (AB+BX) x AD + (AB+BX) x DZ
= AB x AD + BX x AD + AB x DZ + BX x DZ

AB x AD = (AX+XB) x (AZ+ZD)
= (AX+XB) x AZ + (AX+XB) x ZD
= AX x AZ + XB x AZ + AX x ZD + XB x ZD

and that's where i get stuck. how do you go about proving that those 4 vectors are equal to the other 4 vectors? are you suppose to put in arbitary numbers?
 
You may be able to turn this geometric observation into a proof.

Extend ZDY and BXC and mark their intersection as M.
Imagine sliding AB to AX and DC to DM.
Parallelogram AXMD has the same area as ABCD.
Now slide XM to XY and AD to AZ.
Parallelogram AXYZ has the same area as AXMD.
 
Ah indeed robphy!

galois,
to continue with the idea ...
what is BX x AD? (hint : definition of cross product)
what is AB x DZ + BX x DZ ??
(hint : again u can use the definition of cross product here ... )

-- AI
 

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