In trapezoid ABCD CD = 7.5, BC = 4, AD = 8.5. Angles A and B are straight. Find AC.

Draw a perpendicular СН from the vertex С to the AD side. Since CH is perpendicular to AD and AB is perpendicular to AD, then AB = CH.
Consider a triangle CHD: CD = 7.5 – hypotenuse, HD = AD – BC = 8.5 – 4 = 4.5 – leg. Let us find CH by the Pythagorean theorem:
CH = √ (CD ^ 2 – HD ^ 2) = √ (7.5 ^ 2 – 4.5 ^ 2) = √ (56.25 – 20.25) = √36 = 6.
Therefore, AB = 6.
Consider a triangle ABC: AB = 6 and BC = 4 – legs, AC – hypotenuse. By the Pythagorean theorem, we find AC:
AC = √ (AB ^ 2 + BC ^ 2) = √ (6 ^ 2 + 4 ^ 2) = √ (36 + 16) = √52 = √4 * 13 = 2√13.
Answer: AC = 2√13.