Question

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

Answer 1

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

$\dfrac{AO}{CO}=\dfrac{DO}{BO}.$

Then $AO\cdot BO=CO\cdot DO.$

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

$A_{ABO}=\dfrac{1}{2}\cdot AO\cdot BO\cdot \sin\angle AOB.$

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

$A_{COD}=\dfrac{1}{2}\cdot CO\cdot DO\cdot \sin\angle COD.$

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

$A_{COD}=\dfrac{1}{2}\cdot BO\cdot AO\cdot \sin\angle AOB=A_{AOB}=6\ in^2.$

Answer: 6 sq. in.