### Instruction: In triangle $ABC$, $AB = 5$, $BC = 8$, and the length of median $AM$ is 4.  Find $AC$.### Response: I notice that the median $AM$ divides the triangle into two smaller triangles, $ABM$ and $ACM$, that have the same area.
I also know that the area of a triangle is half the product of its base and height, so I can write an equation relating the areas of $ABM$ and $ACM$: $\frac{1}{2} \cdot AB \cdot BM = \frac{1}{2} \cdot AC \cdot CM$.
Simplifying, I get $AB \cdot BM = AC \cdot CM$.
I plug in the given values: $5 \cdot BM = AC \cdot 4$.
I need to find $BM$ and $AC$.
I realize that $BM$ is half of $BC$, since $AM$ is a median.
So $BM = \frac{8}{2} = 4$.
I substitute $BM = 4$ into the equation: $5 \cdot 4 = AC \cdot 4$.
Dividing both sides by 4, I get $AC = 5$.