PABCD = 48, MN-?

The image shows a trapezoid ABCD circumscribed around a circle. The segment MN is the median of the trapezoid. For a trapezoid circumscribed around a circle, the sum of opposite sides is equal. Therefore, AB + CD = AD + BC. Since the perimeter PABCD = AB + BC + CD + AD = 48, we have 2(AB + CD) = 48, which means AB + CD = 24. The length of the median MN of a trapezoid is half the sum of the bases, so MN = (AD + BC)/2. Since AD + BC = AB + CD = 24, MN = 24/2 = 12.