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August 15th, 2013, 05:48 AM  #1 
Newbie Joined: Aug 2013 Posts: 3 Thanks: 0  find the sides and angles of triangle
My question relating to trigonometry. A triangle having altitude magnitude is 40 and its base 100 and base opposite angle is 110 degree, so find the rest of all sides and angles. I attach picture that I draw rough sketch on paint so sorry about sketch. It will clear you the question... please answer my question as quickly as possible.

August 15th, 2013, 07:24 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
When the altitude bisects the base, the angle at B is maximized, and would then be 2arctan(50/40), which is only about 102.68°, so the problem has no solution.

August 15th, 2013, 11:42 AM  #3 
Newbie Joined: Aug 2013 Posts: 3 Thanks: 0  Re: find the sides and angles of triangle
@ skipjack why you take 50 for which method and principle?

August 15th, 2013, 02:26 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the sides and angles of triangle
Is angle A = angle C? Your diagram seems to indicate so.

August 15th, 2013, 02:39 PM  #5 
Member Joined: Aug 2013 Posts: 59 Thanks: 0 
If we call the base point D, then is ?ADB = ?CDB = 90° true?

August 16th, 2013, 02:46 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
I used 100/2 = 50 because I was considering the case where the altitude bisects the base. Using 50  x and 50 + x, one would get arctan((50  x)/40) + arctan((50 + x)/40) = 110°, which has no solution. The LHS is an even function and its maximum value occurs at x = 0.

August 16th, 2013, 03:30 AM  #7 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Re: find the sides and angles of triangle
Even if you don't assume that the altitude hits at the midpoint, there is no solution. Suppose that the altitude divides the side of 100 into two parts ... x on the left and 100x on the right. You then get This simplifies to which has a negative discriminant and thus has no real solutions. 
August 17th, 2013, 04:26 AM  #8 
Newbie Joined: Aug 2013 Posts: 3 Thanks: 0  Re: find the sides and angles of triangle
It's not a midpoint.

August 17th, 2013, 06:17 AM  #9  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  Re: find the sides and angles of triangle Quote:
Highest possible is ~103 degrees; that's with height line dividing AC in 49:51 ratio.  
August 22nd, 2013, 10:25 AM  #10 
Newbie Joined: Jun 2013 Posts: 8 Thanks: 0  Re: find the sides and angles of triangle
You can apply the Generalized Pythagorean Theorem, let’s say that the foot of altitude is H so we have, note BH with x and HC with 100 – x so we have a equation in x with two solutions, both are valid.


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