The diagonals divide the trapezoid into four triangles. Find the area of a triangle adjacent to the larger

| September 11, 2021 | education

The diagonals divide the trapezoid into four triangles. Find the area of a triangle adjacent to the larger base if the areas of the triangles adjacent to the side and the smaller base are 6 and 4, respectively.

Let us construct a perpendicular BH to the AC diagonal.

Consider triangles ABO and BOC, which have a common height BH, then:

Saov = OA * BH / 2 = 6 cm2.

BH = 2 * 6 / OA = 12 / OA.

Svos = OS * BH / 2 = 4 cm2.

BH = 2 * 4 / OC = 8 / OC.

Equate equalities 1 and 2.

12 / OA = 8 / OC.

OS / OA = 8/12 = 2/3.

In triangles BОС and AOD, the angle BОС = AOD as vertical angles, the angle ОBС = ОDА as criss-crossing angles at the intersection of parallel lines BC and AD of the secant BD, then the triangles BОС and AOD are similar in two angles.

OC / OA = 2/3 is the coefficient of similarity of triangles.

Then Svos / Sаod = (2/3) 2 = 4/9.

Saod = Svos * 9/4 = 4 * 9/4 = 9 cm2.

Answer: The area of ​​the triangle at the larger base is 9 cm2.



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