Question

if each side of a triangle is double then find the ratio of area of the new triangle thus formed and the given trinangle

Answer 1

Let a,b,c be the sides of the triangle.

Perimeter 2s = a + b + c
Semi-perimeter, s = (a+b+c)/2
Using Heron's formula:
Area of the triangle A = √s(s−a)(s−b)(s−c)

Now, if the sides are doubled: 2a, 2b, 2c

Let s' be the semi-perimeter.
2s' = 2a + 2b + 2c
s' = a + b + c
or s' = 2s

Area of the triangle, A' = √s′(s′−2a)(s′−2b)(s′−2c)
A' = √(2s)(2s−2a)(2s−2b)(2s−2c)
A' = √24s(s−a)(s−b)(s−c)
A' = 4√s(s−a)(s−b)(s−c)
A' = 4A
A':A = 4:1
Ratio of area of the new triangle and old triangle is 4:1