Question

Meera has a plot in the shape of a parallelogram. She takes a point P in the
interior of her plot and divides it into four parts as shown in the figure. She
then gives triangular plots LPD and APN to her daughter and triangular plots
LPA and DPN to her son.
Has Meera distributed her land equally to both
her children? Justify your answer.
P
А
D
N
MATHS
7 ​

Answer 1

Given : Meera has a plot in the shape of a parallelogram. She takes a point P in the  interior of her plot and divides it into four parts as shown in the figure. She  then gives triangular plots LPD and APN to her daughter and triangular plots  LPA and DPN to her son.

To find : Has Meera distributed her land equally to both  her children

Solution:

Lets draw a line EF passing though  P  such that

EP ⊥ LA  &  FP ⊥ DN

EPF is a straight line as  LA ║ DN

Area of triangle DPN = (1/2) DN *  PF

Area of triangle LPA = (1/2) AL * PE

AL = DN  ( opposite sids)

=> Area of triangle DPN  + Area of triangle LPA  = (1/2) DN *  (PF + PE)  

= (1/2) DN * EF

= (1/2) ( area of parallelogram )

Area of triangle DPN  + Area of triangle LPA  =  (1/2) ( area of parallelogram )

=> Area of triangle APN  + Area of triangle LPD =  area of parallelogram -  (1/2) ( area of parallelogram )  =  (1/2) ( area of parallelogram )

Area of triangle DPN  + Area of triangle LPA  =  Area of triangle APN  + Area of triangle LPD =(1/2) ( area of parallelogram )

Hence YES , Meera has distributed her land equally to both  her children

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