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CTK Exchange
JE
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Jul-23-07, 09:38 AM (EST) |
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"Triangle's area rate of change"
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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? |
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alexb
Charter Member
2057 posts |
Jul-23-07, 09:48 AM (EST) |
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1. "RE: Triangle's area rate of change"
In response to message #0
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>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”ć, 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. |
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JE
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Jul-23-07, 10:12 AM (EST) |
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2. "RE: Triangle's area rate of change"
In response to message #1
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>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 ;)
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alexb
Charter Member
2057 posts |
Jul-23-07, 11:22 AM (EST) |
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3. "RE: Triangle's area rate of change"
In response to message #2
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>>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 Ą 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 https://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml , with varying altitude. >>Area(ABC) <= (b/2)2 ~ cot(Ą/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. |
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