The area of a triangular plot, whose sides are 100m, 105m and 145m, is equal to a rectangular field, whose sides are in the ratio 7:10. The perimeter (in m) of the rectangular field is:

First, let's find the area of the triangular plot using Heron's formula.
The sides are a=100m, b=105m, and c=145m.
The semi-perimeter (s) is:

s=
2
a+b+c
​
=
2
100+105+145
​
=
2
350
​
=175m
Now, compute the area (A
triangle
​
):

A
triangle
​
=
s(s−a)(s−b)(s−c)

​

A
triangle
​
=
175×(175−100)×(175−105)×(175−145)

​

A
triangle
​
=
175×75×70×30

​

A
triangle
​
=
(25×7)×(25×3)×(7×10)×(3×10)

​

A
triangle
​
=
25
2
×7
2
×3
2
×10
2


​
=25×7×3×10=5250m
2

The area of the rectangular field is equal to this triangular area. Let the sides of the rectangle be 7k and 10k:

Area of rectangle=7k×10k=70k
2

70k
2
=5250⟹k
2
=
70
5250
​
=75⟹k=
75

​
=5
3

​

The perimeter of the rectangular field is:

Perimeter=2(Length+Width)=2(10k+7k)=2(17k)=34k
Perimeter=34×5
3

​
=170
3

​
m