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Math Help - length of a side in a rectangle

  1. #1
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    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?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by sanee66 View Post
    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, 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
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  3. #3
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    Hello, sanee66!

    Sorry, your answer makes no sense . . .


    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

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