Posted by **John** on Thursday, August 19, 2010 at 11:13am.

If ABCD is an arbitrary convex quadrilateral, then the area enclosed by ABCD is described by

the formula

(1/2) AC*BD*sin(theta) , where ! is one of the angles formed by the intersecting

diagonals AC and BD. Prove this formula, and explain why it does not matter which of the

four possible angles ! represents.

- Math -
**drwls**, Thursday, August 19, 2010 at 11:28am
Are ! and theta the same angle?

The angles formed by an intersecting pair of lines are pairs of supplementary angles. Since the sine of theta equals the sine of (180 - theta), it makes no difference which angle is used.

Let O be the point where the two diagonals cross. Then

AC*BD = (AO + OC)(BO + OD)

= AO*BO + OC*BO + AO*OD + OC*OD

Those are the pairs of sides that make up the four pieces of the full quadrilateral. When you add then and factor in sin theta and (1/2), you are adding the areas of those four pieces. That gives you the full area.

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