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.
Answer This Question
Related Questions
 Math  Coordinate Geometry  The vertices of a quadrilateral are A(2,3), B(5,4...
 Geometry  The coordinates of the vertices of quadrilateral ABCD are A (8, 8), ...
 maths plse help me..  ABCD ia a parallelogram whose diagonals intersecting at...
 maths  ABCD ia a parallelogram whose diagonals intersecting at E. AC is ...
 geometry  quadrilateral ABCD has right angles at B and D. If ABCD is kite...
 Math  Coordinate Geometry  The vertices of a quadrilateral are A(2,3), B(5,4...
 maths  Given is a quadrilateral ABCD,diagonals AC and BDare such that AO=BO=CO=...
 geometry  Given: QRST is a parallelogram. Prove: QRST is a square. Complete ...
 plllllllllls help math  ABCD is a convex quadrilateral with AC and BD ...
 maths  ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90...
More Related Questions