Answer:
Area of the remaining triangle with the villager is 1243.13 m²
Step-by-step explanation:
Triangle ABC is the triangular plot of a villager shown in the figure attached.
Sarpanch requested the villager to donate land which is 6 m wide and along the side AC which measures 132.8m.
Other sides of the plot has been given as AB = 50m and BC = 123 m.
Now area of this land before donation = [tex]\frac{1}{2}\times {\text{Height}}\times \taxt{Base}[/tex]
= [tex]\frac{1}{2}\times (123)\times (50)[/tex]
= 3075 square meter
After donation of the land the triangle formed is ΔDBE.
In ΔABC,
[tex]tan(ABC)=(\frac{AB}{BC})[/tex]
tan(∠ABC) = [tex]\frac{50}{123}[/tex]
= 0.4065
∠ABC = [tex]tan^{-1}(0.4065)[/tex]
= 22.12°
In ΔEFC,
tanC = [tex]\frac{EF}{CF}[/tex]
0.4065 = [tex]\frac{6}{CF}[/tex]
CF = [tex]\frac{6}{0.4065}[/tex]
CF = 14.76 m
Since DE = AC - (CF + AG)
= 132.8 - (2×14.76)
= 132.8 - 29.52
= 102.48 m
Now in ΔDBE,
sin(∠DEB) = [tex]\frac{BE}{DE}[/tex]
sin(22.12) = [tex]\frac{BE}{102.48}[/tex]
DB = 102.48×0.3765
= 38.59 m
Similarly, cos(22.12) = [tex]\frac{BE}{DE}[/tex]
0.9264 = [tex]\frac{BE}{102.48}[/tex]
BE = 102.48×0.9264
= 94.94m
Now area of ΔDBE = [tex]\frac{1}{2}(DB)(BE)[/tex]
= [tex]\frac{1}{2}(38.59)(94.94)[/tex]
= 1831.87 square meter
Area of remaining triangle with the villager = Area of ΔABC - Area of ΔDBE
= 3075 - 1831.87
= 1243.13 square meter
