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Question Number 16947 by Tinkutara last updated on 28/Jun/17

Through the vertices of the smaller base AB of the trapezoid ABCD two parallel lines are drawn, intersecting the segment CD. These lines and the trapezoid′s diagonals divide it into seven triangles and a pentagon. Show that the area of the pentagon equals the sum of the areas of the three triangles that share a common side with the trapezoid.

$$\mathrm{Through}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the}\:\mathrm{smaller} \\ $$$$\mathrm{base}\:{AB}\:\mathrm{of}\:\mathrm{the}\:\mathrm{trapezoid}\:{ABCD}\:\mathrm{two} \\ $$$$\mathrm{parallel}\:\mathrm{lines}\:\mathrm{are}\:\mathrm{drawn},\:\mathrm{intersecting} \\ $$$$\mathrm{the}\:\mathrm{segment}\:{CD}.\:\mathrm{These}\:\mathrm{lines}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{trapezoid}'\mathrm{s}\:\mathrm{diagonals}\:\mathrm{divide}\:\mathrm{it}\:\mathrm{into} \\ $$$$\mathrm{seven}\:\mathrm{triangles}\:\mathrm{and}\:\mathrm{a}\:\mathrm{pentagon}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pentagon}\:\mathrm{equals} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{areas}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three} \\ $$$$\mathrm{triangles}\:\mathrm{that}\:\mathrm{share}\:\mathrm{a}\:\mathrm{common}\:\mathrm{side} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{trapezoid}. \\ $$

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