PR and QS are the diagonals of the quadrilateral PQRS. Ifθis the acute angle
between PR and QS, show that the area of the quadrilateral PQRS is
1/2 QS x PR sinθ.
Let X be the pt of the intersection of PR and QS
Area of PQRS
= 1/2 (QX)(PX)sinθ+1/2 (RX)(SX) sinθ+1/2(QX)(RX)sinθ+1/2(PX)(SX)sinθ
=1/2sinθ(QXPX+RXSX+QXRX+PXSX)
=1/2sinθ(QX(PX+RX)+SX(PX+RX))
=1/2sinθ(QX+SX)(PX+SX)
=1/2 QS x PR sinθ