Tuesday, February 9, 2010

Problem 424: Triangle, Cevian, Angle, 90 degree, Congruence

Proposed Problem
Click the figure below to see the complete problem 424 about Triangle, Cevian, Angle, 90 degree, Congruence.

Problem 424: Triangle, Cevian, Angle, 90 degree, Congruence.
See also:
Complete Problem 424

Level: High School, SAT Prep, College geometry

Posted by Antonio Gutierrez at 7:18 AM
Labels: , , , ,

4 comments:

  1. AnonymousFebruary 9, 2010 at 8:52 AM

    http://geometri-problemleri.blogspot.com/2010/02/problem-70-ve-cozumu.html

    ReplyDelete
  2. AnonymousFebruary 11, 2010 at 9:56 AM

    Let be m(ACB)=y and m(BAC)=2y
    in triangle ABC: 5y+x=90
    knowing y is knowing x,and 0<5y<90
    with the sine rule in ABD and ABC
    BD=AB.sin2y/cos3y;BC=AB.sin2y/sin3y
    in the right triangle BCD
    BC²+BD²=DC²=AB²
    then sin²(2y)=cos²(3y).sin²(3y)=sin²(6y)/4
    we have the formula: sin(3a)=3sina-a(sina)^3
    with t=2y ,sin²t=1/4
    2y=30
    x=15
    .-.

    ReplyDelete
  3. Sumith PeirisSeptember 10, 2015 at 12:06 AM

    Let E be the mid point of CD and F the mid point of AB. Then E is the centre of right Tr. BCD and so BF = BE.

    Now < ABE = 180 - 2/3<C - 2<C = 180 - 8/3< C. Hence < BFE = < BEF = 4/3<C.

    Hence < FEA = 4/3<C - 2/3<C = 2/3<C = < FAE. So Tr. BEF is equilateral and BE is perpendicular to AC and so < C = 45 and it follows that x = 15

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  4. MarcoDecember 15, 2023 at 9:11 PM

    Let x+2y+3y=90
    x=90-5y
    <BDA=180-(90-5y)-2y=90+3y

    In triangle BAD
    sin(90+3y)/AB=sin2y/BD
    BD/AB=sin2y/cos3y-------(1)

    In triangle BCD
    sin3y=BD/DC-------(2)

    Since AB=CD, (1)=(2)
    sin2y/cos3y=sin3y
    sin2y=sin3ycos3y
    2sin2y=sin6y
    sin6y/sin2y=2
    sin3u/sinu=2, with u=2y
    3-4(sinu)^2=2
    sinu=1/2
    u=30, y=15
    x=90-5y=15

    ReplyDelete

Subscribe to: Post Comments (Atom)