Posted by samuel on Monday, April 18, 2011 at 7:59am.
Two sides of a triangle measure 8 and 15. How many integer values can be assigned to the third side so that all the angles in the triangle are acute?

geometry  MathMate, Monday, April 18, 2011 at 9:02am
The cosine rule gives the formula for the cosine of an angle when the three sides are known:
cos(A)=(B²+C²A²)/(2BC)
where B and C are lengths of sides adjacent to A.
We know that a positive angle is acute when cos(A) is positive, which in turn requires that
B²+C²A²>0 ....(1)
So if all three angles of a triangle have to be acute, we have the two additional requirement analogous to (1) above:
C²+A²B²>0
A²+B²C²>0
If the third side can be chosen such that all three conditions are satisfied, the triangle has acute angles.
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