# Lenght of the Diagonal of a Quadrilateral given its 4 Side Lengths.

• Nov 15th 2009, 12:03 PM
asyed94
Lenght of the Diagonal of a Quadrilateral given its 4 Side Lengths.
I was browsing the AMC 2010 Brochure, and came across problem
10A-12, 12A10. It it possible to solve this problem with the given information? If so, how? Thanks in advance.
• Nov 15th 2009, 01:53 PM
Soroban
Hello, asyed94!

It's much easier thank you imagine . . .

Quote:

In quadrilateral $\displaystyle ABCD\!:\;\;AB = 5,\;BC = 17,\;CD = 5,\;DA = 9.$
And the length of diagonal $\displaystyle BD$ is an integer.
Find the length of $\displaystyle BD.$

Code:

                        A                         o               9    *    *                 *        *       D    *              *         o                  * 5       *      *              *     5 *            *        *     *                    *    *   C o  *  *  *  *  *  *  o B                   17

In $\displaystyle \Delta ABD$, the Triangle Inequality says: .$\displaystyle AB + AD \:> \:BD$

That is: .$\displaystyle 5 + 9 \:>\:BD \quad\Rightarrow\quad BD \:<\:14$

In $\displaystyle \Delta BCD$, the Triangle Inequality says: .$\displaystyle BD + CD \:>\:17$

That is: .$\displaystyle BD + 5 \:>\:17 \quad\Rightarrow\quad BD \:>\:12$

Therefore: $\displaystyle BD$ is an integer between 12 and 14 . . . $\displaystyle BD \,=\, 13.$

• Nov 15th 2009, 02:17 PM
asyed94
Many thanks for the quick and concise reply. This problem had me on the ropes all day :). How were you able to determine the method necessary to solve it?