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

1. ## 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.

2. Hello, asyed94!

It's much easier thank you imagine . . .

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 . . . $BD \,=\, 13.$

3. 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?