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Math Help - [SOLVED] quadrilateral QABR

  1. #1
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    [SOLVED] quadrilateral QABR

    Hi, This question is from the SAT OG Book on page 746 # 12.

    Note: Figure not drawn to scale

    In the figure above, points P, A and B are equally spaced on line L and points P, Q, and R are equally spaced on line M. If PB=4, PR=6, and AQ=4, what is the perimeter of quadrilateral QABR?

    I know from the information that PA=AB=2, and that PQ=QR=3. Therefore I know the sides AB=2, AQ=4, QR=3. How do I find BR?

    Thanks in advance
    Attached Thumbnails Attached Thumbnails [SOLVED] quadrilateral QABR-746.jpg  
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  2. #2
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    Lexington, MA (USA)
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    Hello, fabxx!

    In the figure below, points P, A, B are equally spaced,
    and points P, Q, R are equally spaced.
    If PB=4,\;PR=6,\;AQ=4, what is the perimeter of quadrilateral QABR ?

    Rotate the diagram 90.
    Code:
                          P
                          o
                        *  *
                    3 *     * 2
                    *        *
                Q o * * * * * o A
                *       4      *
           3  *                 * 2
            *                    * 
        R o * * * * * * * * * * * o B
                      ?

    We have: \Delta PBR

    QA joins the midpoints of two sides.

    Hence, QA \parallel RB and QA = \frac{1}{2}(RB)

    Therefore: . RB \:=\:8

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  3. #3
    Member
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    Quote Originally Posted by Soroban View Post
    Hello, fabxx!


    Rotate the diagram 90.
    Code:
                          P
                          o
                        *  *
                    3 *     * 2
                    *        *
                Q o * * * * * o A
                *       4      *
           3  *                 * 2
            *                    * 
        R o * * * * * * * * * * * o B
                      ?

    We have: \Delta PBR

    QA joins the midpoints of two sides.

    Hence, QA \parallel RB and QA = \frac{1}{2}(RB)

    Therefore: . RB \:=\:8

    Hi (: Thanks for your reply, it is correct.

    Can you explain how you got QA = \frac{1}{2}(RB)? And also, does it mean that if QA joins the midpoints of two sides, QA then parallel RB? Does this rule always apply or are there limiations?

    Thanks again!! (:
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  4. #4
    Junior Member
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    Mesnil, Mauritius
    Posts
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    Soroban used the similarity rule.

    The two triangles are similar (same proportions).

    From this rule, you can easily deduce that RB = 2 * QA, since PR=2*PQ.

    Hope that helps.
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