On the midline of the trapezoid ABCD with the bases AD and BC, an arbitrary point E was chosen. Prove that the sum of the areas of the triangles BEC and AED is equal to half the area of the trapezoid
Draw through the point E the height of the trapezoid h.
The height h is divided in half by the point E, since located on the midline, and the middle line divides the sides of the trapezoid in half.
Thus, it turns out that the height of both triangles is equal to h / 2.
The area of a triangle is equal to half the product of height and the base of the triangle.
The area of the trapezoid is equal to the product of half the sum of the bases in height.
SBEC = (h / 2) * BC / 2
SAED = (h / 2) * AD / 2
SBEC + SAFD = (h / 2) * BC / 2 + (h / 2) * AD / 2 = (h / 2) (BC + AD) / 2 = (h * (BC + AD) / 2) / 2 = SABCD / 2
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