Question 1113952: Let N be the number of non-congruent triangles with all the following properties: A. All three sides are integers less than or equal to 15. B. The area of the triangle is an integer. C. At least one angle is 30°, 45°, 60°, 120°, 135°, or 150°.
Compute N.
Answer by ikleyn(52908)   (Show Source):
You can put this solution on YOUR website! .
1. If two adjacent side lengths are integer numbers, then with the concluded angle between them of 30°, 45°, 135° and/or 150°
the opposite side can not have integer length due to the cosine theorem.
It can not even have the length expressed by a rational number.
It follows from the cosine theorem (as I just said) and from the fact that
cos(30°) =  , cos(45°) =  ), cos(135°) =  and cos(150°) = 
are irrational numbers.
So, it kills all the opportunities for angles 30°, 45°, 135° and/or 150°.
2. Regarding the angles 60° and 120°, such triangles also are impossible, since their area
(which is half the product of two adjacent side lengths by sin(60°) or sin(120°))
also can not be integer, because sin(60°) = sin(120°)) = is an irrational number.
Answer. The number of triangles under the question is 0 (zero, ZERO).
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Good problem for the day of April, 1.
April, 1 is an International Fools' day
For further info see this Wikipedia article
https://en.wikipedia.org/wiki/April_Fools%27_Day
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