Proving Parallelogram PQRS & Quadrilateral ABCD: Help Needed

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    Parallelogram Proofs
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Discussion Overview

The discussion revolves around two proofs related to the properties of parallelograms and quadrilaterals, specifically focusing on the area relationships involving triangles within these shapes. The scope includes mathematical reasoning and conceptual clarification regarding area properties and the correct interpretation of problem statements.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about proving that the sum of the areas of triangles TSR and TQP equals half the area of parallelogram PQRS, indicating a need for clarification on properties of parallelograms.
  • Another participant suggests that area formulas are necessary rather than properties of parallelograms to solve the problems.
  • A request for further explanation is made by a participant seeking more detail on the initial claims.
  • One participant challenges the wording of the problems, arguing that the original statements are incorrect and should refer to the areas of the triangles rather than suggesting that triangles can be added or that area can be bisected.
  • Another participant points out that the notation used for triangles is typically understood to refer to the shapes themselves, rather than their areas, suggesting a need for clearer notation when discussing areas.

Areas of Agreement / Disagreement

Participants express disagreement regarding the interpretation of the problem statements and the appropriate terminology to use when discussing areas of triangles versus the triangles themselves. There is no consensus on how to proceed with the proofs, as participants have differing views on the clarity and correctness of the problem wording.

Contextual Notes

Participants note potential limitations in the problem statements, including ambiguous wording and the need for precise definitions regarding area and geometric properties. The discussion highlights the importance of clear communication in mathematical problems.

ruud
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I'm having trouble with two parallelogram proofs

1) PQRS is a parallelogram and T is any point inside the parallelogram. Prove that triangle TSR + triangle TQP = 1/2 parallelogram PQRS

2) ABCD is a quadrilateral whose area is bisected by the diagonal AC. Prove that BD is bisected by AC

I'm stuck on both of them I know that I need to use the Parallel Area Property but I'm unclear or properties of parallelograms. ANy help will be appreciated
 
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You don't need properties of parallelograms. You just need area formulae.
 
Could you please explain further
 
Tell us what you've done so far.
 
First of all, you have stated the problems incorrectly. The problem can't possibly say "Prove that triangle TSR + triangle TQP = 1/2 parallelogram PQRS" because you can't "add" triangles, you can't divide a parallelgram by 2, and two triangles are NOT the same as a parallelgram.

You can, of course, add numbers and divide a number by 2. I'm feel sure that the problem really asked you to show that the sum of the areas of the two triangles is the same as 1/2 the area of the parallelogram. That was why Hurkyl said you need area formulas. Hint: think "base" and "altitude".

"2) ABCD is a quadrilateral whose area is bisected by the diagonal AC. Prove that BD is bisected by AC"

Wow, I hate the wording of that!(And I recognise that it might actually be the way it is worded in your book.) "Area" is a number and cannot be "bisected"! What is meant here is that the diagonal AC divides ABCD into two triangles which each have the same area.
You refer to the "parallelogram area property" but say that you are unsure about the properties. How about quoting the exact statement of the "parallelogram area property" for us? (And once again, think about the formula for area of a triangle. What about two triangles that have the same base and area?)
 
He probably meant 'triangle' as in:

[tex]\Delta TSR + \Delta TQP[/tex]

Which means area
 
Actually, no, [itex]\Delta TSR[/itex] usually means "triangle TSR". To ask for area symbolicaly, you usually prefix it with m, A, or alpha (or even the whole word "Area"), as in [itex]\alpha (\Delta TSR)[/itex]
 
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