1
$\begingroup$

Using vector method -

Show that the lines joining the mid points of the consecutive sides of a quadliteral form a parallelogram.

The link from which i am taking hint suggest to take one point as origin. But i am unable to understand.

As this question involves use of vectors so problem in taking mid points.

Can someone please explain actual procedure to solve these type of questions.

And i have spent so much time to find solution but no link involve use of vectors. So its become more difficult.

  • 1
    Do you know how to find the middle point of a segment, given the coordinates of its end points? Do you know how to find the coordinates of a vector, given the coordinates of its origin and end? – 2017-01-11
  • 0
    I personally don't even like coordinates in this situation; to travel from vertex $A$ to $B$ we use the vector $\overrightarrow{AB}$; to go from $A$ to the midpoint of the edge $AB$ Is half of the above, i.e., $\frac{1}{2} \overrightarrow{AB}$. If you can find a way to write $\overrightarrow{AB} + \overrightarrow{BC}$ as a single vector (draw!), these two ideas are all you need. – 2017-01-11

1 Answers 1

1

We may use the same Lemma (and the same notation) that was crucial in your other question:

If $A,B,C,D$ are four points in the plane, $ABCD$ is a parallelogram iff $A+C=B+D$.

In the present case, we have four generic points in the plane, $P,Q,R,S$.
Then the midpoints of the $PQ,QR,RS,SP$ segments are given, respectively, by: $$ \frac{P+Q}{2},\quad\frac{Q+R}{2},\quad \frac{R+S}{2},\quad\frac{S+P}{2} $$ and is is straightforward to check that $$ \frac{P+Q}{2}+\frac{R+S}{2} = \frac{Q+R}{2}+\frac{S+P}{2} $$ holds since both terms equal $\frac{P+Q+R+S}{2}$. As a consequence:

If $A,B,C,D$ are four distinct points in the plane, the midpoints of the segments $AB,BC,CD,DA$ are the vertices of a parallelogram.

Namely, Varignon's parallelogram, whose sides are parallel to the diagonals of $ABCD$. Remarkably, the area of the Varignon's parallelogram is just half the area of $ABCD$: enter image description here

  • 0
    What is the way or procedure to solve these type of question using vectors? – 2017-01-11
  • 0
    @JohnSr: to do mathematics is very seldom a matter of "procedures". You simply need a solid knowledge of basic geometry and a solid knowledge of basic vector manipulations. With a bit of practice, translating statements from a framework to another will become natural, and you will develop the ability to figure out what is an efficient approach for tackling your problems. – 2017-01-11
  • 0
    Any link to learn geometry because i am weak in this. – 2017-01-11
  • 0
    @JohnSr: thanks to my friend Alexander Bogomolny, this is an *outstanding* resource: http://www.cut-the-knot.org/ – 2017-01-11
  • 0
    And one last thing can you please edit your answer using vectors. – 2017-01-11
  • 0
    I won't do that because I introduced the shortened notation ($A$ in place of $\vec{OA}$, $B-A$ in place of $\vec{AB}$) exactly to avoid typing $\text{\vec}$ too often. Nothing is lost in rigor, so there is no reason to re-introduce the old notation (at least in my humble opinion). – 2017-01-11