How To Prove Trapezoid Median Theorem Using Vectors?

Question

How do you prove the trapezoid median theorem using vectors.
Theorem: The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
(True for ALL trapezoids.)
NEED ANSWER ASAP PLEASE HURRY

Answer 1

This is going to be tough without a picture, so first draw the following trapezoid:



Make the two parallel sides horizontal. Label the bottom points P and Q and label the top points R and S moving counterclockwise. Now let's call your so-called median vector v.



By the vector addition laws, (1/2)SP + PQ + (1/2)QR = v. Trace this on your picture to convince yourself. Start at the left end of v and follow the three vectors on the left hand side of the above equation. You end up at the right end of v.



Similarly, (1/2)PS + SR + (1/2)RQ = v.



Now, just add these two equations together:



PQ + SR = 2v. (Note that (1/2)SP + (1/2)PS = 0.)



From this last equation you get v = (1/2)(PQ + SR), which gives you your desired results.

Answer 2

I have heard people say "elegant" many time. What exactly does that word mean? The most complicated or the simplest proof? Or maybe just the most beautiful proof? I'll give a proof with my most elegant thoughts. ----- Area is simplest. Here's my reasoning (I have not started the proof just yet, but I assume this should work), the area includes the base and height length. We are "given" the height and the base so the median is the only thing we need to find. b1 = length of base on top b2 = length of base on bottom h = length of height m = length of median h(b1 + b2) / 2 = h(b1 + bm) / 4 + h(b2 + bm) / 4 h(b1 + b2) / 2 = h(b1 + b2 + 2bm) / 4 h(b1 + b2) / 2 = h(b1 + b2) / 4 + h(bm) / 2 h(b1 + b2) / 4 = h(bm) / 2 (b1 + b2) / 2 = bm Q.E.D. ----- EDIT: of the EDIT: *O, the area is actually the height multiplied by the median divided by 2 (which I assume is what you meant). Of course, you can also see this in my proof.* Ignore this! I can't believe I messed this up. But still, my proof does show a relationship with height and median and area. ----- I think I found another way to prove it. Make a rectangle from the trapezoid. How does this help? Look, we have formed two triangles (maybe one) both of which has part of the median. We can use similarity to find the length of the cut off parts and then from knowing the remainder, we find the length of the median. The proof will let you understand what I'm saying. Cut the trapezoid from the bottom base to the shorter upper base. As you can see, we have made two triangles containing a small part of the median. b1 = length of top base b2 = length of bottom base h = height The sum of the base of the triangles are b2 - b1 Note that the cut off part of the median is half of the length of the base for each individual triangle. Therefore, total length of the cut off medians is (b2 - b1) / 2 Remember, we have made a rectangle by cutting the trapezoid. The rectangle has the length the same as the median. The length is also b1. Total length of cut off medians + length of the rectangle = length of the median (b2 - b1) / 2 + b1 = (b1 + b2) / 2 = length of median Q.E.D. ----- OK, I thought of another proof. It's basically the same as the second proof except easier to see. Using one of the slants for the trapezoid, you drag it over so that it reaches the other end of the upper base. We have now created a parallelogram. Other then a parallelogram, a triangle is also made. Using similarity and parallelogram facts, we can find the length of the median. ----- Short and insightful is a very *elegant* definition :-)

Answer 3

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