In trapezoid ABCD, point O is the middle of the smaller base BC. Straight lines AO and CD intersect at point E

| May 1, 2021 | education

In trapezoid ABCD, point O is the middle of the smaller base BC. Straight lines AO and CD intersect at point E, AD = 6 dm, BC = 4 dm. Find the ratio EU: CD. Find the ratio of the areas of the triangles EOC and AED.

Since point O, by condition, is the middle of the segment BC, then BO = OC = BC / 2 = 4/2 = 2 cm.

Consider triangles AED and OEC.

The angle at the vertex E of the triangles is common, therefore the angle OEC = AED, the angle EAD is equal to the angle EOC, and the angle EDC is equal to the angle ECO, as the corresponding angles at the intersection of parallel straight lines BC and AD secant AE and ED.

Then the triangles AED and OEC are similar in the first sign of similarity.

Then OC / AD = EC / CD = 2/6 = 1/3.

The coefficient of similarity of triangles is 1/3, and the ratio of the areas of similar triangles is equal to the square of the coefficient of similarity.

Seoc / Saed = (1/3) ^ 2 = 1/9.

Answer: OS / AD = 1/3, Seoc / Saed = 1/9.



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