# length of a side in a rectangle

• Jun 25th 2007, 08:45 PM
sanee66
length of a side in a rectangle
Could someone please look at this? I think the answer is 26 for the length of KL because KL=2(13). Is this right?
• Jun 25th 2007, 09:02 PM
Jhevon
Quote:

Originally Posted by sanee66
Could someone please look at this? I think the answer is 26 for the length of KL because KL=2(13). Is this right?

Why would you assume KL = 2(13)? :confused:

See the slightly modified diagram below.

MJKN is a square, so the diagonal bisects the angles at the corners. so we have 45 degree angles in the positions shown

Remember, $MK = \sqrt {50}$ and $KI = 13$

Now, $\sin 45^{ \circ} = \frac {JK}{MK} = \frac {JK}{ \sqrt {50}}$

$\Rightarrow JK = \sqrt {50} \sin 45^{ \circ} = 5$

Now note that JK = MN = IL

By Pythagoras' Theorem:

$IJ = LK = \sqrt { (IK)^2 + (IL)^2} = \sqrt {13^2 + 5^2} = 12$
• Jun 25th 2007, 09:17 PM
Soroban
Hello, sanee66!

Quote:

In square $JMNK$, diagonal $MK$ measures $\sqrt{50}$
and in rectangle $IJKL$, diagonal $KI$ measures $13$.
What is the length of IJ (or KL)?
Code:

      I                      J          M       * - - - - - - - - - - - * - - - - - *       |  *                  |    __  * |       |      *  13          |  √50 *  |     x |          *          |    *    | x       |              *      |  *      |       |                  *  | *        |       * - - - - - - - - - - - * - - - - - *       L        y            K    x    N

The sides of the square have length $x$.
From Pythagorus, we have: . $x^2 + x^2 \:=\:(\sqrt{50})^2$

Hence: . $2x^2 = 50\quad\Rightarrow\quad x^2 = 25\quad\Rightarrow\quad x = 5$

In right triangle $ILK$, we have: . $IL = 5,\:IK = 13$
From Pythagorus, we have: . $y^2 + 5^2\:=\:13^2\quad\Rightarrow\quad y^2 = 144\quad\Rightarrow\quad y = 12$

Therefore: . $KL \,=\,IJ \,=\,12$