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Solving these equations simultaneously:

The cosine function is positive in Quadrants I and IV, yielding two solutions per period:

Vectors are perpendicular if their dot product is zero:

Let the triangle be a right triangle with legs $ a $ and $ b $, and hypotenuse $ c $. The area $ A $ of the triangle is given by:

The radius $ r $ of the inscribed circle in a right triangle is:

Substitute into the formula for $ r $:

Now, recall that the area can also be expressed in terms of the inradius and semiperimeter $ s $:

Question: A triangle has sides of lengths $ 13 $, $ 14 $, and $ 15 $. What is the length of the shortest altitude?

Let the triangle have sides $ a = 13 $, $ b = 14 $, $ c = 15 $. The shortest altitude corresponds to the longest side, which is $ c = 15 $.

First, compute the area using Herons formula.

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