Question

If each side of a triangle is doubled, then find the ratio of area of new triangle formed and the given triangle.

Answer 1

Let the sides of the triangle be a,b,c respectively.
By Heron`s formula,
Area of a triangle=sqrt(s*(s-a)*(s-b)*(s-c))
A=sqrt(s*(s-a)*(s-b)*(s-c)),where s=(a+b+c)/2

when the sides are doubled,
a`=2a,b`=2b,c`=2c
=>s`=(a`+b`+c`)/2=(a+b+c)=2s

A`=sqrt(s`*(s`-a`)*(s`-b`)*(s`-c`))=sqrt(2s*(2s-2a)*(2s-2b)*(2s-2c))=4*sqrt(s*(s-a)*(s-b)*(s-c))
=>A`=4A
The Area becomes four times the original triangle.