In a trapezoid ABCD with bases AD and BC, the diagonals intersect at point O.

In a trapezoid ABCD with bases AD and BC, the diagonals intersect at point O. Find the diagonal AC if AD = 12cm, BC = 4cm, AO = 9cm.

Let us prove that the ADO triangle is similar to the BCO triangle.

Angle AOD = BOC as vertical angles at the intersection of diagonals BD and AC.

Angle СВО = АDО as criss-crossing angles at the intersection of parallel straight lines АD and ВС of secant ВD.

Then triangle ADO is similar to triangle BCO in two angles.

In similar triangles ADO and BCO BC / AD = OC / OA.

4/12 = OC / 9.

OS = 4 * 9/12 = 3 cm.

Then AC = OA + OC = 9 + 3 = 12 cm.

Answer: The length of the AC diagonal is 12 cm.