A farmer has two triangular fields ∆PQR and ∆PQS inwhich side PQ is common in the given figure PQ=56m,PR=60m,QR=52m,QS=64m - Maths - Heron\s Formula

Question

A farmer has two triangular fields ∆PQR and ∆PQS
inwhich side PQ is common in the given figure
PQ=56m,PR=60m,QR=52m,QS=64m and PS=48m He has marked mid points L
and M on the sides PS and PR respectively By joining LQ and MQ, he
has made a field in the shape of quadrilateral PLQM He grew wheat
in the quadrilateralPLQM, potatoes in ∆MQR and onions in
∆LQS How much area has been used for each crop ?

Abhishek Kr. Jha · Accepted Answer

Dear student,
Here is the solution of your asked query:

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/ck_599a84c9c3406%20jpg">
Here we have;
PQ=56 cm; PR = 60 cm; QR = 52 cm; QS = 64 and PS = 48 cm

Considering △PQR we have;semi-perimeter of △PQR;s=PQ+QR+PR2=56+52+602=84 cmSo using Heron's formula we have;Area of △PQR=8484-5684-5284-60                             = 84×28×32×24                            = 7×12×7×4×16×2×2×12                            = 7×12×2×4×2                            = 1344 cm2Now considering △PQS we have;semi-perimeter of △PQS;s'=PQ+QS+PS2=56+64+482=84 cmSo using Heron's formula we have;Area of △PQS=8484-5684-6484-48                             = 84×28×20×36                            = 7×2×2×3×7×4×4×5×6×6                            = 7×2×4×63×5                            =33615                            = 1301
32 cm2Now L is the midpoint of PS, so we have;area of △PQL=area of △LQS=12area of △PQS=12×1301
32 = 650
66 cm2Also M is the midpoint of PR, we have;area of △PQM=area of △QMR=12area of △PQR=12×1344 = 672 cm2Also area of quadrilateral PQLM= area of △PQL+area of △PQM=650
66+672=1322 66 cm2
Regards