In trapezoid ABCD, point M lies on the side of AB. O- the point of intersection of the diagonal BD and the segment CM

| April 30, 2021 | education

In trapezoid ABCD, point M lies on the side of AB. O- the point of intersection of the diagonal BD and the segment CM. Find the area of the trackball BOC if BM = 2 AM, CO = 5 OM and the area of the triangle COD is 1.

Draw from point M a straight line MP parallel to the bases of the trapezoid and intersecting the diagonal BD with point E.

Triangles ВOС and MOE are similar in two angles, then OK / OH = OС / OM = 5 * X / 1 * X.

Then the length of the segment KH = 6 * X.

Let us extend the segment KH to the intersection with the base AD.

Then KН / НL = ВM / AM = 2/1.

NL = KН / 2 = 3 * X. Then KL = KН + НL = 9 * X.

Triangles BOC and ВСD have a common side ВС.

Then the ratio of their areas is equal to the ratio of their heights.

Svos / Svsd = KL / KO = 5 * X / 9 * X = 5/9.

9 * Svos = 5 * Svsd.

Svsd = Svsd – Ssod = Svsd – 1. (multiply by 5)

5 * Svos = 5 * Svsd – 5.

5 * Swax = 9 * Swax – 5.

4 * Swax = 5.

Swax = 5/4 = 1.25 cm2.

Answer: The area of ​​the VOC triangle is 1.25 cm2.



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