1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Geometry tetrahedron problem

  1. Dec 24, 2015 #1
    1. The problem statement, all variables and given/known data

    Volume of tetrahedron T.ABC = V
    Point P is on the middle of TA, Q is on the expansion of AB making AQ = 2AB
    A shape is made through PQ which is parallel to BC so that it cuts the tetahedron into 2 pieces.
    What is the volume of the biggest piece?

    3. The attempt at a solution

    I sketch the problem


    Then, I have no idea how to continue
  2. jcsd
  3. Dec 24, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    Hi Terry,

    I am missing something ! : 2. Relevant equations
    For this part of the template, you could think of this or this
    Deleting (even just a part of) the template irritates the spirits that watch over us, so you really don't want to do that. A litte googling usually helps if your notes or textbook don't have anything useful. And in this case the second link might have given you an idea...

    I notice you've drawn PM parallalel to AC instead of NM parallel to BC. Perhaps you could clarify ?

    And, for the record, I don't have the answer, so we still need help. :frown:

  4. Dec 24, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Nevertheless, you need to make an attempt; those are the PF rules.
  5. Dec 24, 2015 #4
    I'm sorry. Now, I want to edit the thread but it's disabled now :frown:
    Okay, I just add them in this post

    - Relevant equations

    V = Base Area * height

    - My Attempts

    I see the answer in the book, but it's very messy and I don't understand.


    Please help

    I don't understand why it is TN/NB = TM/MC (I think it should be TN/TB = TM/TC)
    Then, TP/TA*TN/TB*TM/TC comes out which gets me more confused
    I think the book has lots of typos, but I'm pretty sure the book's got the idea

    (The book is written in Bahasa, this is the translation :
    P is the center point of TA,
    Q is on the expansion of AB => AQ = 2AB
    The shape through PQ//BC that cuts the tetrahedron into two pieces is)
    Last edited: Dec 24, 2015
  6. Dec 25, 2015 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Likewise, TM/TC=1/(1+MC/TM).
    So if TN/TB=TM/TC then TN/NB=TM/MC.
  7. Dec 25, 2015 #6
    Why is it 2/1 ? And, how does TP/TA*TN/TB*TM/TC come from? I really get nothing from the book answer
    Last edited: Dec 25, 2015
  8. Dec 25, 2015 #7


    User Avatar
    Science Advisor
    Homework Helper

    I don't know if it is the easiest way, but the theorem of Menelaus, applied to the triangle TAB and the line PNQ, shows that TN/NB = 2.
  9. Dec 25, 2015 #8


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Consider the centroid of triangle TAQ.
    The volume of a tetrahedron Is 1/6 of the triple scalar product of the vectors representing ithree sides adjacent to one vertex. So if you keep the angles fixed and vary the lengths, it is proportional to the product of the three lengths.
  10. Dec 25, 2015 #9
    Isn't the volume of tetrahedron is a^3/(6√2) where a is the length of the edge??
    But, how to know that the angle is fixed?? The shape is a bit different from the big initial tetrahedron I think.
  11. Dec 25, 2015 #10


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    For a regular tetrahedron, yes, but this is not.
    We're keeping the three angles at T fixed.
    Think of T as the origin. The volume of TABC is one sixth the triple scalar product of vectors TA, TB, TC. Likewise, the volume of TMNP is one sixth the triple scalar product of the vectors TM, TN, TP.
    The triple scalar product of three vectors is the product of the magnitudes of the vectors, multiplied by a function of the angles between them.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Geometry tetrahedron problem Date
Circle Geometry with an Intersecting Line Mar 10, 2018
Triangle inscribed in a circle Jan 4, 2018
Radius of insphere in a Tetrahedron Mar 7, 2015