Monday

April 21, 2014

April 21, 2014

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

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 11:28amAre ! 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.

**Related Questions**

maths - Given is a quadrilateral ABCD,diagonals AC and BDare such that AO=BO=CO=...

geometry - ABCD is a parallelogram. Let C′ be a point on AC extended such ...

geometry - Quadrilateral ABCD is a rhombus with diagonals AC and BD. if angle M ...

plllllllllls help math - ABCD is a convex quadrilateral with AC and BD ...

geometry - quadrilateral ABCD has right angles at B and D. If ABCD is kite-...

Math - The bases of trapezoid ABCD are AB and CD. Let P be the intersection of ...

heeeeeelp math - In a tetrahedron ABCD, the lengths of AB, AC, and BD are 6, 10...

helllllppp math - In a tetrahedron ABCD, the lengths of AB, AC, and BD are 6, 10...

maths - I have an illustration in my book of a quadrilateral inside a circle. ...

maths - ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90...