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Thread: Triangle. Find QT

  1. #1
    Junior Member
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    Triangle. Find QT

    Okay so Ive figured out half of this question so far, and I need help finding out the rest.

    PQRS is an isosceles trapezoid with PS=QR=17 and PQ = 16 and SR = 32. Point R is on SV with RV = 16 and point Q is on the hypotenue ST of right triangle STV. The length of QT is:
    a) 26 b) 3 x root of 89 c) 4 x root of 57 d) 3 x root of 93 e) 39

    So Ive figured out that the height of the trapezoid is 15
    and that QS = root of 801
    can someone help me find QT with this information given and explain why ?
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    Last edited by foreverbrokenpromises; May 23rd 2009 at 07:21 PM. Reason: Forgot to Attach diagram
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  2. #2
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    Hello, foreverbrokenpromises!

    I can't get your diagram to show up
    . . but I think I've figured it out.


    $\displaystyle PQRS$ is an isosceles trapezoid with: .$\displaystyle PS=QR=17,\;PQ = 16,\;SR = 32$

    $\displaystyle SR$ is extended to $\displaystyle V$ so that $\displaystyle RV = 16.$

    Point $\displaystyle Q$ is on the hypotenue $\displaystyle ST$ of right triangle $\displaystyle STV.$

    The length of $\displaystyle QT$ is:

    . . $\displaystyle (a)\;26\qquad (b)\;3\sqrt{89} \qquad (c)\;4\sqrt{57} \qquad (d)\;3\sqrt{93}\qquad (e)\;39$


    So I've figured out that the height of the trapezoid is 15 . Right!
    . . and that $\displaystyle QS \:=\: \sqrt{801} \:=\:3\sqrt{89}$ . Yes!

    Can someone help me find $\displaystyle QT$ with this information given and explain why?
    Code:
                                              o T
                                           *  |
                                        *     |
                                     *        |
                                  *           |
                P    16     Q  *              |
                o-----------o                 |
               /:        *  :\                |
              / :     *     : \               |
             /  :  *        :  \              |
            /   *         15:   \             |
           / *  :           :    \            |
          o-----+-----------+-----o - - - - - o
          S  8  A    16     B  8  R    16     V

    Draw $\displaystyle PA \perp SR$
    Draw $\displaystyle QB \perp SR$
    Then: .$\displaystyle SA = 8,\;AB = 16,\;BR = 8$

    We see that: .$\displaystyle \Delta QBS \sim \Delta TVS$

    Hence: .$\displaystyle \frac{ST}{SQ} \:=\:\frac{SV}{SB} \quad\Rightarrow\quad \frac{ST}{SQ} \:=\:\frac{48}{24} \quad\Rightarrow\quad ST \:=\:2\!\cdot\!QS $

    Divide by 2: . $\displaystyle \tfrac{1}{2}ST \:=\:QS$


    Since $\displaystyle QT = \tfrac{1}{2}ST$, we have: .$\displaystyle QT \:=\:QS \:=\:3\sqrt{89} $ . . . answer (b)

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  3. #3
    Junior Member mathhomework's Avatar
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    Here is my answer.

    Some how my answer is not one of the choice. I am pretty sure of my answer, but I could be wrong. Please let me know.

    QT=34


    I know what I did wrong! Sorry!
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