Problem
Find the length of the segment indicated. Round your answer to the nearest tenth if necessary. 12.5 14.4 12.2
Solution
Step 1 ：Draw the triangle and label the points of interest.
Step 2 ：Notice that the triangle is isosceles with $AB = AC = 10$ and $BC = 12$.
Step 3 ：Since $ABC$ is isosceles, $AD$ is a median and an altitude.
Step 4 ：Calculate $BD = DC = \frac{1}{2} \cdot BC = 6$.
Step 5 ：Recognize the $3:4:5$ Pythagorean triple and find $AD = 8$.
Step 6 ：Calculate the area of $ABC$ using $\frac{1}{2} \cdot AD \cdot BC = \frac{1}{2} \cdot 8 \cdot 12 = 48$.
Step 7 ：Find $BE$ using the area: $\frac{1}{2} \cdot AC \cdot BE = 48$, so $BE = 9.6$.
Step 8 ：By symmetry, $CF = 9.6$.
Step 9 ：Add the lengths of the altitudes: $9.6 + 9.6 + 8 = \boxed{27.2}$.
