 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Angle ABC (Posted on 2006-02-25) In a triangle ABC, D is the midpoint of BC. Join AD. Angle ADB = 45 degree and angle ACB = 30 degree. Find angle ABC.

 No Solution Yet Submitted by akash Rating: 2.3000 (10 votes) Comments: ( Back to comment list | You must be logged in to post comments.) Another approach | Comment 10 of 13 | A similar approach to that of vije...

Extend CB to O, so that AO is perpendicular to CO.

WLOG, letting CD = DB = 1,
cot 30� = (2 + BO)/AO = sqrt(3),
cot 45� = (1 + BO)/AO = 1.
Hence BO/AO = cot ABO = 2 - sqrt(3).
(The three terms form an arithmetic progression.)

Now we can either note(!) that cot 75� = 2 - sqrt(3), or note that we can use the addition formula for cotangents:

cot 75� = cot (45� + 30�)
= (cot 45� cot 30� - 1)/(cot 45� + cot 30�)
= (sqrt(3) - 1)/(1 + sqrt(3))
= 2 - sqrt(3)

Either way, angle ABO = 75�, and so angle ABC = 105�.
Edited on February 26, 2006, 1:00 pm
 Posted by Nick Hobson on 2006-02-26 12:57:07 Please log in:

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