posted by John on .
If ABCD is an arbitrary convex quadrilateral, then the area enclosed by ABCD is described by
(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.
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.