Wednesday, February 18, 2009

Problem 15: Triangle, Cevian, Angles, Congruence

Proposed Problem
Problem 15: Triangle, Cevian, Angles, Congruence.

See complete Problem 15 at:
http://gogeometry.com/problem/problem015.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

Posted by Antonio Gutierrez at 5:06 PM
Labels: , , ,

11 comments:

  1. AjitMarch 17, 2009 at 12:54 AM

    Extend BA to point E such that AE=AD. Now,
    AB+AD = AB+AE = BE. But BC=AB+AD. Hence BE=BC. In triangles EBD & CDB, AD is common, the included angles ABD & DBC are each = 30 and BE=BC which makes the triangles congruent. Thus angle AED = x and angle ADE also =x since AD=AE by construction while angle BAD=x+x=2x and sngle ADB=x+30. Triangle ABD now gives, 30+2x+30+x=180 so x=40 deg.
    Ajit: ajitathle@gmail.com

    ReplyDelete
  2. AnonymousJuly 2, 2010 at 2:56 AM

    http://ahmetelmas.files.wordpress.com/2010/05/cozumlu-ornekler.pdf

    ReplyDelete
  3. bae deok rak(bdr@korea.com)October 4, 2010 at 8:05 PM

    Let E be a point on BC with BE=BA. Since BA=BE and ang(ABE)=60 deg,triangle BAE is an Equilateral triangle and DA=DE=EC. Hence we see that ang(DCE)=x=ang(CDE) and ang(DEA)=x/2 and hence 2x=x/2+60, that is , x=40 degree.

    ReplyDelete
  4. UnknownApril 15, 2012 at 4:47 AM

    The solution is uploaded to the following link:

    https://docs.google.com/open?id=0B6XXCq92fLJJMk52SXpZdlZFRFU

    ReplyDelete
  5. Sumith PeirisJuly 20, 2015 at 4:50 AM

    Extend BA to F such that BF = BC. Hence Tr. FBC is equilateral and both Tr.s FAD & FDC are isoceles since BD is the perpendicular bisector of FC
    So from Tr. FDC 60-x= 60-A/2 and so x= A/2, which can only happen when A= 80 and x=40 since B= 60

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  6. geoclidOctober 15, 2016 at 9:01 AM

    https://www.youtube.com/watch?v=Rg1AM58ShnE

    ReplyDelete
  7. AnonymousJuly 28, 2017 at 8:21 PM

    Construct point E, which lies on AB extended, such that AE=AD. Join AE and DE.
    Triangle BED and BCD are congruent (SAS)
    Hence <BED=x
    Since AE=AD,
    <ADE=x
    So <BAD=2x
    By angle sum of triangle in ABC,
    x=(180-30-30)/3
    x=40

    ReplyDelete
  8. rv.littlemanFebruary 22, 2019 at 9:24 AM

    See original solution on this video

    ReplyDelete
  9. rv.littlemanFebruary 25, 2019 at 10:57 AM

    See original solution on this video

    ReplyDelete
  10. GregMarch 4, 2019 at 5:58 AM

    Great proof by rv !

    ReplyDelete
  11. Ercan CemMarch 18, 2019 at 7:38 AM

    Construct point E on the left of AC such that AE=AB. Verily, EBC is isosceles, and so is BAE. Now, 3x=120 --> x=40.

    ReplyDelete

Subscribe to: Post Comments (Atom)