A right triangle has legs of length 6 and 8 and a hypotenuse of length 10. What is the length of the altitude drawn from the right angle…

Compute the triangle's area two ways. With the legs as base and height, the area is $\frac{1}{2}(6)(8) = 24$. Using the hypotenuse of length 10 as the base and the altitude $h$ as the height, the area is $\frac{1}{2}(10)h = 5h$. Setting these equal gives $5h = 24$, so $h = 4.8 = \frac{24}{5}$. Common wrong answers: 24 is the area, not the altitude; 5 is half the hypotenuse, which would be the median to the hypotenuse; and 7 or 3.5 come from averaging or subtracting the legs. The altitude to the hypotenuse is always shorter than either leg.