geometry

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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 -

    The cosine rule gives the formula for the cosine of an angle when the three sides are known:
    cos(A)=(B²+C²-A²)/(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+C²-A²>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+A²-B²>0
    A²+B²-C²>0

    If the third side can be chosen such that all three conditions are satisfied, the triangle has acute angles.

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