Parallelogram problem: use coord's to prove in general

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Jamers328

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Sep 20, 2007
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I don't know if any of you are familiar with Geometer's Sketchpad, but we had to do this activity using that program. It is not necessary that you know the program to do the problem, it would just make more sense because that is exactly what I needed to do with it. Anyway, you can help regardless I'm sure. :D

Activity:
Construct a parallelogram ABCD, with A at the origin and B on the positive x-axis. Draw the diagonal AC, and construct point P on this diagonal. Then draw segments PB and PD. This creates four small triangles. Construct the interior of each triangle (select the three vertices and go to the Construct menu), and measure the area of each triangle. Vary the location of P, and form a conjecture about these areas. Then use coordinates to prove your conjecture in general.

The ONLY part I really need help with is the underlined part, the very last sentence. I got everything up to that point. I do not know how to prove the conjecture (which is basically that there are two pairs of triangles that have the same area) using coordinates.

Thank you for your time!
 
Re: Parallelogram problem

Possibly, I understand what you are after. Suppose the coordinates of the various points are as follows:

A(0,0); B(6,0); C(8,6); P(4,4)

You can determine the area of triangle ABP because the base (AB) is 6 units and the altitude is the segment from P perpendicular to AB. The altitude is 4 units. Therefore the area of ABP is (1/2)(6)(4) = 12 square units.
The area of BCP is the area of ABC minus the area of ABP. We know the area of ABP. The area of ABC is its base (AB=6) times its height (distance from C perpendicular to AB = 6) times 1/2 which is 18 square units.
Area of ABC minus area of ABP= 18 - 12 = 6 square units. Etc.

Hope that helps?
 
Thank you! That makes sense and I think my teacher may be looking for a different way to do it, but I can't figure it out, so I am using your method. Thanks a bunch! :D
 
Jamers328 said:
I don't know if any of you are familiar with Geometer's Sketchpad, but we had to do this activity using that program. It is not necessary that you know the program to do the problem, it would just make more sense because that is exactly what I needed to do with it. Anyway, you can help regardless I'm sure. :D

Activity:
Construct a parallelogram ABCD, with A at the origin and B on the positive x-axis. Draw the diagonal AC, and construct point P on this diagonal. Then draw segments PB and PD. This creates four small triangles. Construct the interior of each triangle (select the three vertices and go to the Construct menu), and measure the area of each triangle. Vary the location of P, and form a conjecture about these areas. Then use coordinates to prove your conjecture in general.

The ONLY part I really need help with is the underlined part, the very last sentence. I got everything up to that point. I do not know how to prove the conjecture (which is basically that there are two pairs of triangles that have the same area) using coordinates.

Thank you for your time!
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