In trapezoid ABCD, the diagonals intersect at point E. The area of triangle ABE is 72, the area of triangle CDE is 50. Find the area of trapezoid ABCD.
Triangles AEB and DEC are similar in two angles. Angle DEC = AEB as vertical angles, angle EDC = EBA as criss-cross.
Since the ratio of the areas of similar triangles is equal to the square of the coefficient of similarity of triangles, then:
Sde / Save = K ^ 2 = 50/72 = 25/32.
K = 5/6.
Consider triangles ADE and CDE in which the height DH is common, then the ratio of the areas of these triangles is equal to the ratio of their bases.
Since CE / AE = K = 5/6, then:
Sde / Sade = 5/6.
Sade = 6 * Sde / 5 = 6 * 50/5 = 60 cm2.
By the trapezoid property Sade = Sall = 60 cm2.
Then Strap = 50 + 72 + 60 + 60 = 242 cm2.
Answer: The area of the trapezoid is 242 cm2.
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