• Sep 2nd 2009, 03:18 PM
messianic
Here are the questions:

1.)

In the triangle ABC, let M and N be the midpoints of AB and AC respectively. Show that vector MN is equal to one-half of vector BC. Conclude that the line segment joining the midpoints of two sides of a triangle is parallel to the third side. How are their lengths related?

2.)

Use vectors to prove that the midpoints of the four sides of an arbitrary quadrilateral are the vertices of a parallelogram.

Thanks A Lot
• Sep 3rd 2009, 01:28 AM
red_dog
1) $\displaystyle \overrightarrow{MN}=\overrightarrow{MA}+\overright arrow{AN}=\frac{1}{2}\overrightarrow{BA}+\frac{1}{ 2}\overrightarrow{AC}=$

$\displaystyle =\frac{1}{2}(\overrightarrow{BA}+\overrightarrow{A C})=\frac{1}{2}\overrightarrow{BC}$

The vectors $\displaystyle \overrightarrow{MN}$ and $\displaystyle \overrightarrow{BC}$ are colinear, then $\displaystyle MN\parallel BC$

$\displaystyle |\overrightarrow{MN}|=\left|\frac{1}{2}\overrighta rrow{BC}\right|\Rightarrow MN=\frac{1}{2}BC$

2) Use the first problem in the triangles formed by three vertices of the parallelogram.
• Sep 3rd 2009, 01:13 PM
messianic

I tried using your logic for the 2nd problem but I keep getting the wrong answers. I think im messing up the vector addition

Could you please give a brief explanation for the 2nd problem?

Thanks!
• Sep 4th 2009, 02:29 AM
red_dog
Let ABCD be the parallelogram, M the midoint of AB, N the midoint of BC, P the midoint of CD, Q the midpoint of DA.

Use the problem 1 in the triangle ABC with the points M, N, then in the triangle CDA with the points P, Q.

You'll get that MN and PQ are parallel to AC and their length is half of AC. So MNPQ is a parallelogram.
• Sep 4th 2009, 04:15 AM
aidan
Quote:

Originally Posted by red_dog
Let ABCD be the parallelogram, M the midoint of AB, N the midoint of BC, P the midoint of CD, Q the midpoint of DA.

Use the problem 1 in the triangle ABC with the points M, N, then in the triangle CDA with the points P, Q.

You'll get that MN and PQ are parallel to AC and their length is half of AC. So MNPQ is a parallelogram.

This just provided insight to the solution of a problem.
Thanks.