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.
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.
1 answer
1 answer