In trapezoid ABCD (AD and BC base), the diagonals intersect at point O, AD = 12 cm

In trapezoid ABCD (AD and BC base), the diagonals intersect at point O, AD = 12 cm, BC = 4 cm.Find the area of triangle BOC if the area of triangle AOD is 45 cm ^ 2

Let us prove that triangle AOD is similar to triangle BOC.

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

Angle ВСО = DАО as criss-crossing angles at the intersection of parallel straight lines ВС and АD of secant АС. Then the triangles BOC and AOD are similar in two angles.

Let’s determine the coefficient of similarity of triangles. K = BC / AD = 4/12 = 1/3.

The ratio of the areas of similar triangles is equal to the squared coefficient of their similarity.

Swax / Saod = 1/9.

Swax = Saod / 9 = 45/9 = 5 cm2.

Answer: The area of the ВOС triangle is 5 cm2.