Question

QUESTION:Mr. Ramesh is a farmer by profession. PQRS is a piece of land in parallelogram shape, which belongs to Mr Ramesh. He wants to leave the yellow coloured area free and wants to grow wheat and maize in the remaining field giving equal area to both the crops. His friend Manu suggests to divide the area PTRM into two congruent triangles and the grow both the cropsin them. But Mr Ramesh is not sure that both the triangles PTR and RMP are congruent and hence have equal areas.Can youhelp him by proving that 훥푃푇푅≅훥푅푀푃?Also, prove that 푅푇∥푃푀and RT=PM.

Answer 1

Given :  PQRS is a parallelogram , PT = MR  & PT || MT

To Find : Show that Δ PTR  ≅   ΔRMP

Solution:

Draw TA ⊥ PR

& MB ⊥ PR

Compare ΔPTA  &  Δ RMB

PT = RM   given

∠PAT = ∠RBM  = 90°

∠APT = ∠BRM   ( alternate interior angle )

=> Δ  ΔPTA  ≅  Δ RMB

=> TA = MB

PA = RB

area of ΔPTR = (1/2)  * PR * TA

area of ΔRMP= (1/2)  * PR * MB

TA = MB

Hence area of ΔPTR = area of ΔRMP

PA = RB

=> PA + AB = RB + AB

=> PB = RA

Compare ΔPBM  &  Δ RAT

PB = RA

∠PBM = ∠RAT = 90°

TA = MB    

=> ΔPBM  ≅ Δ RAT

=> PM  = RT

Comparing  Δ PTR  &   ΔRMP

PT = RM

RT = PM

PR =  RP  ( common )

=> Δ PTR  ≅   ΔRMP

Congruent triangle have equal area

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