Results 1 to 5 of 5

Math Help - Midpoint of square's diagonal - proof

  1. #1
    Member
    Joined
    Mar 2011
    Posts
    92

    Midpoint of square's diagonal - proof

    Hi All,

    I am not able to solve this geometry problem correctly. Visually it looks plausible but how do you go about proving it.

    Midpoint of square's diagonal - proof-m_center_square.gif

    In the diagram, ABCD is a square, EFG is an isosceles triangle, Angle EFG = 90, and M is midpoint of EG.

    (a) Show that Angle EFA = Angle FGB.

    (b) Explain why Triangle AFE, and BGF are congruent.

    (c) If AC is drawn, explain why M is the midpoint of AC.

    Also one more question! Is there a way to describe such geometry problems in latex? I did the the attachment in mspaint, was wondering if there is a better way? Thanks for your help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,454
    Thanks
    1868
    The crucial point is that right angle, EFG. Because of that the two angles AFE and GFB are "complementary"- the measures of the three angles, AFE, EFG, and GFB, add up to 180 degrees because they form a straight line and since EFG is a right angle, AFE and GFB add to 90 degrees.

    And then, look at right triangles EAF and FGB. Angle EFA is complementary to AEF and FGB is complementary to GFB. That, together with the fact that AFE and GFB are complementary, shows that EFA and FGB are congruent.

    For (b) you need the additional given information, that "EFG is an isosceles triangle". That is, sides EF and FG are congurent. Then use "ASA".

    For (c), notice that drawing horizontal and vertical lines through M divides the square into 4 smaller but identical squares.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2011
    Posts
    92
    Thanks @HallsofIvy. That clears up the fog quite a bit!

    Regarding (a) is this a corollary like ASA, SSS, etc of similar/rt triangles, ie:- if non right angles of right triangles are complementary it is similar?

    Regarding (c) can you please clarify how to show that the points A, C and M are collinear?

    Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,561
    Thanks
    785
    Also one more question! Is there a way to describe such geometry problems in latex? I did the the attachment in mspaint, was wondering if there is a better way?
    Check out this FAQ. On this forum, you can use the picture environment. See an example in this thread. Elsewhere, PGF/TikZ is a very powerful graphic language for TeX. Its author also wrote an awesome Beamer class for creating presentations. However, the learning curve for TikZ is pretty steep.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2011
    Posts
    92
    Thanks @emakarav. I'll check it out.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving for side of square given diagonal
    Posted in the Geometry Forum
    Replies: 4
    Last Post: December 19th 2012, 06:28 PM
  2. Replies: 4
    Last Post: June 7th 2010, 05:45 PM
  3. Replies: 0
    Last Post: October 15th 2008, 05:05 AM
  4. [SOLVED] Exact value of a Square's diagonal
    Posted in the Geometry Forum
    Replies: 3
    Last Post: March 9th 2008, 06:11 PM
  5. Diagonal matrix proof
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 6th 2008, 01:45 PM

Search Tags


/mathhelpforum @mathhelpforum