Trapezium

Geometry Level 3

The A B C D ABCD is trapezium, where A B ∣ ∣ C D AB\mid\mid CD and A B = 2 ∗ C D AB=2*CD . The F F point divides the A D AD side to two equal parts. The A C , B F , B D AC, BF, BD segments form the colored triangles.

Which area is larger, the red or the blue?

They are equal The blue area The red area Can't be determined

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Áron Bán-Szabó
Jul 11, 2017

Complete the trapezoid to the A B G D ABGD parallelogram. [ B F D ] = 1 2 [ A B D ] = 1 4 [ A B G D ] [BFD]=\dfrac{1}{2}[ABD]=\dfrac{1}{4}[ABGD] [ D A C ] = 1 2 [ D A G ] = 1 4 [ A B G D ] [DAC]=\dfrac{1}{2}[DAG]=\dfrac{1}{4}[ABGD] The equations above can be seen in the figure. As a consequence, [ B F D ] = [ A D C ] [BFD]=[ADC] . By subtract [ E F G D ] [EFGD] we get red area = sum of the areas of the blue triangles \text{red area}=\text{sum of the areas of the blue triangles}

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