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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.