# Proof verification

Printable View

• March 2nd 2011, 04:15 PM
scounged
Proof verification
I was recently presented with this problem:

$\mbox{Let ABCD be a parallelogram, where the diagonal BD is equal to the side AB. Show that}$

$\displaystyle{\frac{\mid{AD}\mid}{\mid{AC}\mid} < \frac{2}{3}{$

$\mbox{where}\mid{AD}\mid\mbox{and}\mid{AC}\mid\mbo x{are the lengths of the sides.}$

I solved this problem by first drawing up the parallelogram, like this:
Attachment 21025

Then I added a few extra lines, like this:
Attachment 21026
adding the height h and a point P.

After that I set a condition that $h>0$ and stated that $\mid{BD}\mid=\mid{AB}\mid=\mid{DC}\mid$ and also the fact that
$\mid{DP}\mid=\frac{\mid{AD}\mid}{2}$, which I'm too lazy to prove right now, as I'm still struggling with latex.

After this I used the pythagorean theorem to show that ${\mid{AC}\mid}^2=(1.5{\mid{AD}\mid})^2+h^2$ which means that $\mid{AC}\mid=\sqrt{(1.5{\mid{AD}\mid})^2+h^2}$

As an effect, this means that $\mid{AC}\mid>\sqrt{(1.5{\mid{AD}\mid})^2}$, as $\sqrt{(1.5{\mid{AD}\mid})^2+h^2}$ clearly is bigger than $\sqrt{(1.5{\mid{AD}\mid})^2}$

Then, I just made myself an inequality.

$\mid{AC}\mid>1.5{\mid{AD}\mid}$

$\displaystyle{\frac{\mid{AC}\mid}{1.5\mid{AD}\mid} >1}$

$\displaystyle{\frac{1}{1.5\mid{AD}\mid}>\frac{1}{\ mid{AC}\mid}}$

$\displaystyle{\frac{\mid{AD}\mid}{1.5\mid{AD}\mid} >\frac{\mid{AD}\mid}{\mid{AC}\mid}}$

And then finally

$\displaystyle{\frac{\mid{AD}\mid}{\mid{AC}\mid}<\f rac{2}{3}}$, which I was supposed to show.

My question is if I've done everything right, as the book in which I found the problem solves it with trigonometry, and the section for the problem was a section about trigonometry. If my solution is incorrect I need to know why.
• March 7th 2011, 07:24 AM
abhishekkgp
It's correct. :)
but there is a geometrical proof i got.
Drop perpendicular from B to AD and call the foot of the perpendicular as K. Let AC meet BD at L.
then |AK|=|KD| and |AL|=|LC|.
Let AL and BK meet at G. G is the centroid since AL and BK are medians.
To prove: |AD|/|AC| < 2/3
that is, |AK|/|AL| < 2/3.
Now |AK|<|AG| since AG is the hypotenuse of triangle-AGK.(AGK is a right angled triangle).
so |AK|/|AL| < |AG|/|AL|.
we know that |AG|/|AL|=2/3 from elementary geometry because G is the centroid. which proves the result.