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Math Help - vector

  1. #1
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    vector

    In a parallelogram OABC , it is given OA = a , OC = c , PB = 1/3AB and BQ = 1/3 BC . OQ and AB are extended to meet at S while CB abd OP are extended to meet at R
    Express (a) BS in terms of c
    (b) BR in terms of a

    (c) Show that RS is parallel to AC
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  2. #2
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    Quote Originally Posted by mathaddict View Post
    In a parallelogram OABC , it is given OA = a , OC = c , PB = 1/3AB and BQ = 1/3 BC . OQ and AB are extended to meet at S while CB abd OP are extended to meet at R
    Express (a) BS in terms of c
    (b) BR in terms of a

    (c) Show that RS is parallel to AC
    a) Express BS in terms of c.

    Given: OC = c, and OA = a

    So, BC = OA = a

    BQ = (1/3)BC = a/3
    CQ = BC -BQ = 2a/3

    angle BQS = angle CQO .........vertical angles.
    angle BSQ = angle COQ ........alternate interior angles.
    Therefore, triangle BQS is similar to triangle CQO .....two angles of each triangle are equal each to each.

    Similar triangles are proportional, by proportion,
    OC/CQ = BS/BQ
    c /(2a/3) = BS /(a/3)
    Cross multiply,
    c(a/3) = (BS)(2a/3)
    c = 2BS
    BS = c/2 --------------answer.

    -------------------------------------------------
    a) Express BR in terms of a.

    Given: OC = c, and OA = a

    So, AB = OC = c

    PB = (1/3)AB = c/3
    AP = AB -PB = 2c/3

    angle RPB = angle OPA .........vertical angles.
    angle BRP = angle AOP ........alternate interior angles.
    Therefore, triangle BRP is similar to triangle AOP.
    By proportion,
    OA/AP = BR/PB
    a /(2c/3) = BR /(c/3)
    Cross multiply,
    a(c/3) = (BR)(2c/3)
    a = 2BR
    BR = a/2 --------------answer.

    ------------------------------------------
    (c) Show that RS is parallel to AC

    angle RBS = angle CBA ...........vertical angles

    BR/CB = BS/AB
    (a/2)/a = (c/2)/c
    1/2 = 1/2
    True

    Hence, triangle RBS is similar to triangle CBA.
    And so, angle ACB = angle SRB
    Therefore, RS is parallell to AC .....with RC as the transversal line, a pair of their alternate interior angles are congruent.
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