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Triangle geometry find a side length

  1. Jun 10, 2012 #1
    1. The problem statement, all variables and given/known data
    In triangle ABC, the median from C meets AB at D. Through M, the midpoint of CD, line AM is drawn meeting CB at P. If CP=4, find CB.


    2. Relevant equations



    3. The attempt at a solution
    I constructed this drawing on GSP and found CB to be 12. I'm trying to show similarity between some triangles in the drawing but can't find any. I would like to know how to solve this without GSP. any ideas??
     
  2. jcsd
  3. Jun 10, 2012 #2

    tiny-tim

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    Hi Wildcat! :smile:

    Hint: areas. :wink:
     
  4. Jun 10, 2012 #3
    Ok, I don't see where I can calculate any areas with the information I have unless I'm missing something. Will I need to construct another segment?
     
  5. Jun 11, 2012 #4

    tiny-tim

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    Hi Wildcat! :smile:

    (just got up :zzz:)
    Yes.

    Divide the triangle into triangles, call two of the unequal areas "p" and "q", and add them all up. :smile:
     
  6. Jun 11, 2012 #5

    VietDao29

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    Hi Wildcat,

    Apart from the area method, there's still another way to tackle this problem. It's to use mid-segment of a triangle (it's the line segment that connects the two midpoints of any 2 sides of a triangle).

    There are 2 theorems about mid-segment you should remember is:
    Given [itex]\Delta ABC[/itex]
    • If M, and N are respectively the midpoints of AB, and AC then [itex]MN = \frac{1}{2}BC[/itex], and [itex]MN // BC[/itex].
      This theorem means that the mid-segment of a triangle is parallel to the opposite side, and is half of it.​
    • If a line passes through the midpoint of one side, and is parallel to the second side, then it also passes through the midpoint of the other side.

    -------------------------------

    So back to your problem,

    Let d be a line that passes through D, and parallel to AM, it intersects BC at Q. Now, look at the 2 theorems above, what conclusion can you draw about P, and Q?

    Hint: Look closely at the 2 triangles [itex]\Delta ABP[/itex], and [itex]\Delta CDQ[/itex]

    Cheers,
     
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