# Thread: length of a side in a rectangle

1. ## 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?

2. 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)?

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

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

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

Now note that JK = MN = IL

By Pythagoras' Theorem:

$\displaystyle IJ = LK = \sqrt { (IK)^2 + (IL)^2} = \sqrt {13^2 + 5^2} = 12$

3. Hello, sanee66!

In square $\displaystyle JMNK$, diagonal $\displaystyle MK$ measures $\displaystyle \sqrt{50}$
and in rectangle $\displaystyle IJKL$, diagonal $\displaystyle KI$ measures $\displaystyle 13$.
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 $\displaystyle x$.
From Pythagorus, we have: .$\displaystyle x^2 + x^2 \:=\:(\sqrt{50})^2$

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

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

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