When x is infinitely close to 180, then this triangle is infinitely close to a straight line, and BC is infinitely close to AB+AC=8.

This is equivalent to from left to right, B, A and C are on the same straight line, BA=5, AC=3, then the length of the midpoint between point A and point BC is 1. If you don't understand, draw it yourself.

So the minimum BC is 1.

When x is infinitely close to 0, A, C and B are on the same straight line from left to right. Similarly, if AC = 3 and AB = 5, then the length of the midpoint between point A and point BC is AB-0.5*BC=5- 1=4.

So the range of BC midpoint is (1, 4)