Question

PQRS is a quadrilateral in which the diagonals PR and QS intersect at L.
Show that ar (Triangle PLS)×ar(Triangle QLR)=ar(Triangle PLS)×ar(Triangle RLS)
please send quickly.

Answer 1

The products of areas of the two opposite triangles 

We know that ∠PLS = ∠QLR   and  ∠PLQ = ∠RLS
Also : ∠PLQ = 180° - ∠PLS   Hence Sin ∠PLQ = Sin ∠PLS
Also:  ∠RLS = 180° - ∠QLR   Hence  Sin ∠RLS = Sin ∠QLR

Area of triangle PLS = 1/2 * LP * LS * Sin ∠PLS
Area of triangle QLR = 1/2 * LQ * LR * Sin QLR = 1/2* LQ*LR* Sin∠PLS
Product :  1/4 * LP*LS*LQ*LR* Sin²∠PLS

Similarly the product of areas of triangles PLQ and RLS:
  = 1/4 * LP*LS*LQ*LR * Sin² ∠PLQ
  = 1/4 * LP* LS*LQ * LR * sin²∠PLS

Hence both products are equal.