play and relax: games for kids games
  Cut the knot: learn to enjoy mathematics
A math books store at a unique math study site. Learn to enjoy mathematics.
Google
Web CTK
Try our no ads browsing

Best sites for teachers
Sites for teachers
Sites for parents
Terms of use
Awards

Interactive Activities
CTK Exchange
CTK Insights - a blog

Games & Puzzles
What Is What
Arithmetic/Algebra
Geometry
Probability
Outline Mathematics
Make an Identity
Book Reviews
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
Visual Illusions
My Logo
Math Poll
Cut The Knot!
MSET99 Talk
Other Math sites
Front Page
Movie shortcuts
Personal info
Privacy Policy

Guest book
News sites

Recommend this site

Best sites for teachers
Sites for teachers
Sites for parents

Education & Parenting

Manifesto: what CTK is about Buying a book is a commitment to learning Table of content Try our no ads browsing Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

The 80-80-20 Triangle Problem, Solution #3

 
  Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 50°. Point E is on side AB such that BCE = 60°. Find the measure of CED.

Solution

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

This is a trigonometric solution (Solution 2 from [Knop]).

 

Denote the unknown angle CED as x. Then CDE = 160° - x.

Apply the Law of Sines in ΔCDE:

  CE : CD = sin (160° - x) : sin x.

And in ΔBCE,

  CE : BC = sin 80° : sin 40° = 2cos 40°.

For, sin 2α = 2sin α · cos α.

Since ΔBCD is isosceles, CD = BC, which allows us to write a trigonometric equation:

  sin (160° - x) : sin x = sin 80° : sin 40°.

This we are going to solve presently.

Using sin(180° - α) = sin(α),

  sin (20° + x) = 2cos 40° · sin x = 2cos (60° - 20°) · sin x.

Using the addition formula for sine and the one for cosine,

  sin 20° · cos x + cos 20° · sin x = cos 20° · sin x + √3sin 20° · sin x,

which simplifies to

  sin 20° · cos x = √3sin 20° · sin x,

or,

  ctg x = √3.

From which x = 30°.

Reference

  1. C. Knop, Nine Solutions to One Problem, Kvant, 1993, no 6.

Copyright © 1996-2008 Alexander Bogomolny

29711605Page copy protected against web site content infringement by Copyscape


Search:
Keywords:


Latest on CTK Exchange
try this puzzle ?/?? + ?/?? + ?/? ...
Posted by albert1950
2 messages
03:40 PM, Aug-26-08

Numbers raised to the power of 0
Posted by Chris Tolley
20 messages
12:17 PM, Aug-25-08

Arbelos : 1) Geometrical Construc ...
Posted by Sundar Krishnan
12 messages
06:29 AM, Aug-12-08

concerning pi
Posted by Lloyd Marks
4 messages
08:25 AM, Aug-22-08

Triangles With Equal Area
Posted by Bui Quang Tuan
5 messages
07:20 PM, Aug-26-08

Coxeter Introduction to Geometry
Posted by WiZaRd
1 messages
09:15 AM, Aug-23-08

site questions
Posted by madisonv
2 messages
04:24 PM, Aug-26-08