In a parallelogram PQRS, L and M are the mid-points of QR and RS respectively. Prove that :

ar( ΔPLM) = 3/8 ar( IIgm PQRS).

Please help. :)

Question

In a parallelogram PQRS, L and M are the mid-points of QR and RS
respectively Prove that :
ar( ΔPLM) = 3/8 ar( IIgm PQRS)
Please help :)

Lalit Mehra · Accepted Answer

PQRS is a parallelogram L and M are the mid point of sides QR and RS respectively Join SQ and draw LN || PQ In Î”SRQ, L and M are the mid point of sides QR and RS respectively $∴ LM | | SQ and LM = \frac{1}{2} SQ (Mid point theorem)$ It can be proved that, $area (Δ LMR) = \frac{1}{4} \times area (Δ SRQ) (1)$ We know that, a diagonal of a parallelogram divides it into two triangle of equal area $∴ area (Δ SRQ) = \frac{1}{2} × area (| |^{gm} PQRS) \Rightarrow area (Δ LMR) = \frac{1}{4} \times \frac{1}{2} area (| |^{gm} PQRS) [Using (1)] \Rightarrow area (Δ LMR) = \frac{1}{8} area (| |^{gm} PQRS) (2)$ area (|| gm PQLN) = area (|| gm SRLN) [Parallelograms on equal base (LQ = LR) and between the same parallels are equal in area] $∴ area (| |^{gm} PQLN) = area (| |^{gm} SRLN) = \frac{1}{2} area (| |^{gm} PQRS) (3) area (Δ PQL) = \frac{1}{2} \times area (| |^{gm} PQLN) = \frac{1}{2} \times \frac{1}{2} area (| |^{gm} PQRS) = \frac{1}{4} area (| |^{gm} PQRS) (4)$ Similarly, $area (Δ PMS) = \frac{1}{4} area (| |^{gm} PQRS) (5)$ area (Î”PLM) = area (|| gm PQRS) â€“ [ area (Î”LMR) + area (Î”PQL) + area (Î”PMS)] $= area (| |^{gm} PQRS) - [\frac{1}{8} area (| |^{gm} PQRS) + \frac{1}{4} area (| |^{gm} PQRS) + \frac{1}{4} area (| |^{gm} PQRS)] = area (| |^{gm} PQRS) - (\frac{1 + 2 + 2}{8}) area (| |^{gm} PQRS) = area (| |^{gm} PQRS) - \frac{5}{8} area (| |^{gm} PQRS) = (\frac{8 - 5}{8}) area (| |^{gm} PQRS) = \frac{3}{8} area (| |^{gm} PQRS) ∴ area (Δ PLM) = \frac{3}{8} area (| |^{gm} PQRS)$

In a parallelogram PQRS, L and M are the mid-points of QR and RS respectively. Prove that : ar( ΔPLM) = 3/8 ar( IIgm PQRS). Please help. :)

In a parallelogram PQRS, L and M are the mid-points of QR and RS respectively. Prove that :

ar( ΔPLM) = 3/8 ar( IIgm PQRS).

Please help. :)