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In trapezoid ABCD with legs AB and CD , point O is the intersection of the diagonals. If AABO=6 in2, find ACOD.

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1. Point O (the point of intersection of diagonals) divides two diagonals AC and BD into parts that are proportional:


(AO)/(CO)=(DO)/(BO).

Then
AO\cdot BO=CO\cdot DO.

2. Consider triangle ABO. The area of this triangle is


A_(ABO)=(1)/(2)\cdot AO\cdot BO\cdot \sin\angle AOB.

3. Consider triangle COD. The area of this triangle is


A_(COD)=(1)/(2)\cdot CO\cdot DO\cdot \sin\angle COD.

Since
AO\cdot BO=CO\cdot DO and angles AOB and COD are congruent as vertical angles, then


A_(COD)=(1)/(2)\cdot BO\cdot AO\cdot \sin\angle AOB=A_(AOB)=6\ in^2.

Answer: 6 sq. in.


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