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

asyed94 · #1 $Old$ November 15th, 2009, 01:03 PM

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.

Soroban · #2 $Old$ November 15th, 2009, 02:53 PM

Hello, asyed94!

It's much easier thank you imagine . . .

Quote:

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

Code:

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

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

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

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

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

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

asyed94 · #3 $Old$ November 15th, 2009, 03:17 PM

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?