#0, Triangle's area rate of change
Posted by JE on Jul-23-07 at 09:38 AM
I'm trying to find a relation between the area of an scalene triangle and its obtuse angle while keeping the base constant. So, lets say I have a triangle_ABC with its obtuse angle_ABC = 100, if one increases the ange_ABC while keeping the base (segment_AC) constant, the area gets smaller. What can I use to state the relation between the angle_ABC and the area?
#1, RE: Triangle's area rate of change
Posted by alexb on Jul-23-07 at 09:48 AM
In response to message #0
>I'm trying to find a relation between the area of an scalene
>triangle and its obtuse angle while keeping the base
>constant. So, lets say I have a triangle_ABC with its
>obtuse angle_ABC = 100, if one increases the ange_ABC while
>keeping the base (segment_AC) constant, the area gets
>smaller. The area does not solely depend on the angle ABC. For example, you can reduce the area to zero without changing the angle, but only the altitude. The vertex B in this case remains on a circular arc ABC while approaching one of the ends (A or C.) >What can I use to state the relation between the angle_ABC
>and the area? Unless you take into account some other parameter, like the altitude from B, the best you can do is establish an inequality for the are, say, Area(ABC) <= (b/2)2 × cot(β/2) / 2, where b is the base and β is the angle ABC. The estimate is exact for isosceles triangles.
#2, RE: Triangle's area rate of change
Posted by JE on Jul-23-07 at 10:12 AM
In response to message #1
>The area does not solely depend on the angle ABC.
if you put a constraint and let the segment AC constant, there's a straight relationship between β and the altitude. If the altitude decreases, β increases. Can this relation be formalized?>Area(ABC) <= (b/2)2 cot(β/2) / 2,
can you give more insights on how you got this? Thanks! I really appreciate it ;)
#3, RE: Triangle's area rate of change
Posted by alexb on Jul-23-07 at 11:22 AM
In response to message #2
>>The area does not solely depend on the angle ABC.
>if you put a constraint and let the segment AC constant,
>there's a straight relationship between and the
>altitude. No, this is incorrect. AC is your base and has been fixed from the outset. For a given angle ABC point B may be anywhere on a circular arc, see http://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml , with varying altitude. >>Area(ABC) <= (b/2)2 ~ cot(/2) / 2,
>can you give more insights on how you got this? Draw a picture and try some trigonometry and do not forget the triangle ABC is supposed to be isosceles. Do try it.
|