All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Measure that angle IV (Posted on 2004-11-02)
Triangle ABC is isosceles with AB=AC. Point D is on side AB such that angle BCD is 70 degrees. Point E is on side AC such that angle EBC is 60 degrees. Angle ABE equals 20 degrees, and angle DCE equals 10 degrees.

 See The Solution Submitted by Brian Smith Rating: 3.6667 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Generalization | Comment 15 of 21 |
` `
`Let triangle ABC be isosceles with AB=AC. Angle BACequals 20 degrees. Let points D and E be on sides ABand AC repectively such that angle BCD is greater thanangle CBE. What integer measures ( in degrees ) canangles BCD and CBE take such that the measure ( indegrees ) of angle EDC is an integer.`
`Let b, c, and d be the measures of angles CBE, BCD,and EDC respectively. Let F be the intersection ofBE and CD. Applying the sine rule to triangles`
`  EDF:   EF*sin(b+c-d) = DF*sin(d)`
`  BDF:   BF*sin(80-b) = DF*sin(100-c)`
`  CEF:   CF*sin(80-c) = EF*sin(100-b)`
`  BCF:   BF*sin(b) = CF*sin(c)`
`Eliminating the lengths from these equations gives`
`     sin(d)       sin(b)*sin(80-c)*sin(100-c)     P  ------------ = ----------------------------- = ---   sin(b+c-d)     sin(c)*sin(80-b)*sin(100-b)     Q`
`solving for d we get`
`              P*sin(b+c)  tan(d) = ----------------            Q + P*cos(b+c)`
`I wrote a small program in Perl and got`
`    BCD  CBE  EDC  -----------------     50   20   10     50   40   30     60   30   10     60   50   30   <--   Langley's Problem     65   25    5     65   60   40     70   50   10     70   60   20   <--   This Problem`
`There might be round off errors in the program, soit would be nice to have a synthetic proof ordisproof for each of these cases.`
` `
` `
` `

 Posted by Bractals on 2004-11-06 14:28:55

 Search: Search body:
Forums (0)