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A problem on geometry.

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Let ABC be an equilateral triangle with side 10 cm and let P be a point inside the triangle at a distance of 2 cm from the side AB. Given --> AB = BC = CA = 10cm and PD = 2 cm. If PD, PE, PF are the perpendiculars to the three sides, find out the sum, PD + PE + PF.

    Here is the image : http://postimage.org/image/1lt6hjgw4/ [Broken]

    2. Relevant equations

    I am not sure which equation is most relevant being a geometry question.

    3. The attempt at a solution

    I tried using similarity of triangles and Pythagoras theorem and trigonometry. I tried messing up all these things but failed to get the solution.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 11, 2011 #2

    ehild

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    Try to express to area of the triangle with PE, PD and PF and with the side of the equilateral triangle.

    ehild
     
  4. Oct 11, 2011 #3
    This is how I worked : http://postimage.org/image/1wf8ya2sk/ [Broken]
    Am I correct ?

    I found these equations :

    x + 2*sqrt(3)/3 = r+y
    x+y+r = 10
    x-r-y= - 2*sqrt(3)/3

    Am I Correct ?
     
    Last edited by a moderator: May 5, 2017
  5. Oct 11, 2011 #4

    ehild

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    I can not follow you. Why did you note by x both the upper and lower parts on the left-hand side of the triangle? The problem can be certainly solved by using similar triangles, but it would be complicated.

    Why do you not try the area method? What are the areas of the coloured triangles in terms of PD, PE, PF? What is the area of the big triangle?


    ehild
     

    Attached Files:

    Last edited: Oct 12, 2011
  6. Oct 12, 2011 #5
    OK , I get it .

    Area of green Δ = 1/2 x 10 x PE = 5PE
    Area of yellow Δ = 1/2 x 10 x PF = 5PF
    Area of light blue Δ = 1/2 x 10 x PD = 5PD

    Area of big Δ = sqrt(3)/4 x 100 = 25 sqrt(3)

    5(PE + PF + PD) = 25 sqrt(3)
    PE + PF + PD = 8.66025 cm approx.

    Now , I can't understand why they gave me the length of PD. There was no need of it.

    Thanks for the efforts.
    :smile:
     
  7. Oct 12, 2011 #6

    ehild

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    To confuse the student. Really. Those evil teachers (including myself) do such things on purpose.:devil:

    ehild
     
  8. Oct 13, 2011 #7
    Really ? Well that's strange and evil for sure devil-devil-monster-evil-smiley-emoticon-000132-large.gif

    Anyways , thanks for your efforts !
     
  9. Oct 13, 2011 #8

    ehild

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    This is a very good picture of me. Have we met before? :rofl:

    ehild
     
  10. Oct 14, 2011 #9
    No.:rofl:

    That is just a smiley which I think you liked.
    :smile:
    Thanks again.
     
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