Proposed Problem

See complete Problem 13 at:

http://gogeometry.com/problem/problem013.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

## Wednesday, February 18, 2009

### Problem 13: Triangle, Angles, Congruence, Cevian

Labels:
angle,
cevian,
congruence,
triangle

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Solucion:

ReplyDeletehttp://forogeometras.com/index.php?topic=884.0

http://img25.imageshack.us/img25/7149/rsolmarzo8.jpg

trazar DE//AB

ReplyDeleteel angulo EDC=4x

Los triangulos ABD y DCE son congruentes

entonces ABD=3x

en el triangulo ABC: 3x+4x+3x+5x=180

x=12º

atte:tino!!!

Dear Tino,

ReplyDeleteThe answer is not correct.

Why triangles ABD and DCE are congruent?

We've CD/sin(5x)=BC/sin(8x) and AB/sin(3x)=BC/sin(4x). Thus, sin(5x)/sin(8x)=sin(3x)/sin(4x) or

ReplyDeletesin(5x)/2sin(4x)cos(4x)=sin(3x)/sin(4x) or

sin(5x)/2cos(4x)=sin(3x)

By inspection if x=10 then sin(5x)=cos(4x) and sin(3x)=1/2. Thus, x=10 is clearly a solution to the equation.

Ajit: ajitathle@gmail.com

This comment has been removed by a blog administrator.

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

ReplyDeleteSolution by cristian, my partnert >:)

ReplyDeletehttp://youtu.be/neHBBXZiaRI

Go Go Geometry!

Geo-Greetings ^^!

https://www.youtube.com/watch?v=-i4RlIl4B0s

ReplyDelete1. Draw E on AB such that BE = BD and F on CD such that DF = BD also. This forms 2 isosceles triangles BDE (with 2 6x angles) and BDF (with 2 4x angles).

ReplyDelete2. By angle chasing <BFA = 4x = <BAD so tr. ABF is isosceles also and

BF = AB = CD.

3. By angle chasing <BDC = <EBF = 180 - 8x so tr BEF = tr BCD by SAS.

4. So <BEF = <CBD = 5x and by angle subtraction <DEF = x

then also <BFE = <BCD = 3x and by angle subtraction <EFD = x also.

5. So tr. EDF is also isosceles and DE = DF = BD = BE.

6. This implies tr. BDE is actually equilateral and 6x = 60. X = 10.

Btw: compare this construction to problem #9. They are the same.

Delete