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Math Help - Triangle PQR...

  1. #1
    Junior Member
    Joined
    May 2009
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    Triangle PQR...

    In triangle PQR, <p = 30 degrees
    <Q = 80 degrees
    <R = 70 degreees
    QH is an altitude and RM is a median and H is on the line PR,
    the measure of <MHP is:

    a) 15 degrees or 20 degrees or 30 degrees or 40 degrees or 45 degreess ?

    diagram is attatched
    Attached Files Attached Files
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  2. #2
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
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    Multiple hellos. It is 30 degrees.

    Solution: (All angular measurements in degrees)
    \angle P = 30 degrees
    \angle Q = 80 degrees
    \angle R = 70 degrees
    QH is an altitude and RM is a median and H is on the line PR

    If QH is an altitude, then \angle QHR and \angle QHP are 90 .
    Filling in relevant angles angles: \angle PHQ=60,

    Area of \triangle PQR=\frac{PRQH}{2}

    A median splits a triangle into two triangles equal in area.
    Construct altitude MO. (dotted line in diagram, forgot to label O)

    Two similar triangles are discovered, \triangle MOP and \triangle PHQ<br />
(as MOand QHare parallel)
    Calculate area of \triangle PMR=\frac{MOPR}{2}=\frac{PRQH}{4}
    (Because a median splits a triangle into two triangles equal in area.)

    Hence,  MO=\frac{QH}{2}
    Which means that PO=\frac{PH}{2} and PM=\frac{PQ}{2}

    Thus OH=PH-\frac{PH}{2}=\frac{PH}{2}
    Hence \triangle MPO and  \triangle MOH are identical
    Hence \angle MHP=30
    Attached Thumbnails Attached Thumbnails Triangle PQR...-attached.bmp  
    Last edited by I-Think; May 25th 2009 at 06:05 PM.
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