Results 1 to 2 of 2

Thread: Triangle PQR...

  1. #1
    Junior Member
    May 2009

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member I-Think's Avatar
    Apr 2009
    Multiple hellos. It is 30 degrees.

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

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

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

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

    Two similar triangles are discovered, $\displaystyle \triangle MOP$ and $\displaystyle \triangle PHQ
    $ (as $\displaystyle MO$and $\displaystyle QH$are parallel)
    Calculate area of $\displaystyle \triangle PMR=\frac{MOPR}{2}=\frac{PRQH}{4}$
    (Because a median splits a triangle into two triangles equal in area.)

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

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

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Apr 23rd 2011, 08:10 AM
  2. Replies: 3
    Last Post: Apr 30th 2009, 07:41 AM
  3. Replies: 1
    Last Post: Oct 28th 2008, 07:02 PM
  4. Replies: 7
    Last Post: Jul 19th 2008, 06:53 AM
  5. Replies: 27
    Last Post: Apr 27th 2008, 10:36 AM

Search Tags

/mathhelpforum @mathhelpforum