Prove the given theorem using vectors:
The midpoints of the sides of quadrilateral are the vertices of a parallelogram.
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If you do, you've probably also seen that we can use lower-case letters to stand for these vectors: , and so on.
We can then use the vector law of addition to write down vectors representing each of the sides of the quadrilateral. For instance, we can say:So we get the vector representing the side as:
This is an extremely important and useful result. Can you use it to find vectors representing the other three sides?
Another vital piece of information is the the position vector of the mid-point of a line. Suppose, for instance, that is the mid-point of the line segment . Then , because is just half-way from to .
So we can write down its position vector (from our origin ) as:You'll see that this is just the 'average' of the position vectors and . That makes sense, because is the 'average position' of and . OK?
, using the result above
, when we simplify.
In the same way if we call N the mid-point of AD, its position vector is:(I'll leave it to you to check this out.)
If we now use the first result above we can get the vector that represents the line joing as:Now that's just one-half of the vector (again using that important result above).
Well now, if you do the same thing for the mid-points of BC and CD, you will get exactly the same result: . (I'll leave that to you.)
Can you see that this proves that these four mid-points form a parallelogram?