# Parallelogram Proofs

1. Feb 25, 2004

### ruud

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

2. Feb 25, 2004

### Hurkyl

Staff Emeritus
You don't need properties of parallelograms. You just need area formulae.

3. Feb 25, 2004

### ruud

4. Feb 25, 2004

### Hurkyl

Staff Emeritus
Tell us what you've done so far.

5. Feb 26, 2004

### HallsofIvy

Staff Emeritus
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?)

6. Feb 26, 2004

### Warr

He probably meant 'triangle' as in:

$$\Delta TSR + \Delta TQP$$

Which means area

7. Feb 26, 2004

### Hurkyl

Staff Emeritus
Actually, no, $\Delta TSR$ 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 $\alpha (\Delta TSR)$

Last edited: Feb 26, 2004