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:

`(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.