Proving questions!!? Watch

Edamame
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#1
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I really suck at proving questions. How do I prove these?

(i) The line that joins the midpoints of two sides of a triangle is parallel to the third side.

(ii) Two diagonals of a square are perpendicular.

I think you're meant to use vectors... Sigh...
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EvenStevens
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I often have the urge to write "BECAUSE THEY JUST ARE! OKAY!"

I think you can prove them by drawing them on a set of axis and finding out the equations of the lines and just matching the gradients but vectors sounds ... less hassle.
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BibbleJW
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Really, there are a number of ways to prove something:
Exhaustive proof - basically go through all possible ways of something happening.
Algebraic proof - If it's true for a,b, and c, then it is true for all cases.
Inductive proof - Prove for n=1, and for n+1, given that it is true for n, and so it is true for everything.

Algebraic is probably best for these examples.
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Edamame
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I have no idea what you guys are talking about...

E.g. for the triangle. I let OPQ be the triangle.

The OP + PQ + QO = 0

I will let the midpoints of OP and OQ be P' and Q' respectively.

Now am I supposed to prove that the angle OQ'P' and OQP is the same for parallel? How would I do that??
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DFranklin
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(Original post by Edamame)
I have no idea what you guys are talking about...

E.g. for the triangle. I let OPQ be the triangle.

The OP + PQ + QO = 0

I will let the midpoints of OP and OQ be P' and Q' respectively.

Now am I supposed to prove that the angle OQ'P' and OQP is the same for parallel? How would I do that??
In this case, the simplest thing is to show explicitly that PQ is a multiple of P'Q'.
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JohnnySPal
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The square one should be easyish, really. If you just define the points A,B,C,D given by (0,0), (0,1), (1,0) and (1,1) to be the corners of the squarte, you can start joining corners to corners using simple vector equations (like y=mx+c), and take dot products of the gradient vectors?

Spoiler:
Show

The lines would be t(1,1) and (0,1) + t(1,-1)
(for t in the reals)

Then take dot product of gradients:
(1,1).(1,-1) = 1-1 = 0

Dot product being zero means perpendicular.


Note that because any square in the plance can somehow be gotten from the square we defined to have the corners A,B,C,D - and most notably because dilating the square won't affect the gradient of the lines that join the corners (easy to see?) we can use the above without loss of generality.
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