In trapezoid ABCD (BC || AD), point M divides the diagonal AC in a ratio of 1: 2, counting from vertex A.

| February 18, 2021 | education

In trapezoid ABCD (BC || AD), point M divides the diagonal AC in a ratio of 1: 2, counting from vertex A. Find the ratio of the area of triangle AMD to the area of trapezoid ABCD.

Let’s draw the height of the trapezoid KН through point M.

The triangles KMС and AMН are similar in two angles, the angles KMС and AMН are vertical, and the MAН and KСM are crosswise. Then AM / CM = HH / KM = 1/2.

KM = 2 * MН.

Then KН = MН + KM = 3 * MН.

The area of ​​the AMD triangle will be equal to:

Samd = (AD * MН) / 2.

The area of ​​the trapezoid is:

Savsd = (BC + AD) * KН / 2 = (BC + AD) * 3 * MН / 2.

((AD * MН) / 2) / (ВС + AD) * 3 * MН/ 2 = AD / (BC + AD) * 3 = 1/3 * (AD / (BC + AD).

Samd / Savsd = (1/3) * (AD / (BC + AD).

The area ratio is equal to one third of the ratio of the larger base to the sum of the bases of the trapezoid.

Answer: Samd / Savsd = (1/3) * (AD / (BC + AD).



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