Question

How many times area is changed whenall sides of a triangle are doubled.​

Answer 1

Answer:

In a triangle let sides are a ,b, c , and (a+b+c)/2=s say , then area of the triangle =√{s(s-a)(s-b)(s-c)}

now if the sides are doubled then sides will be 2a, 2b, 2c , and (2a+2b+2c)/2=(a+b+c)=2s

then area of the triangle

=√{2s(2s-2a)(2s-2b)(2s-2c)}

=√{2×2×2×2×s(s-a)(s-b)(s-c)}

=4√{s(s-a)(s-b)(s-c)}

=4×area of smaller triangle .

Another procedure :-

Let ABC is a triangle , sides AB, BC, CA . Let D,E,F are the middle points of AB, BC, CA respectively. We joined DE,EF,FD . As D and F are the middle points of AB and AC , DF=(1/2) BC . So in triangle ADF all sided are half of respective sides of the big triangle ABC. In this way we can prove all four triangles ADF, BDE, DEF, and CEF are made of half of three sides of triangle ABC. So those four triangles are of equal sizes are made from total area of triangle ABC and sides of each small triangles are half of sides of the big triangle.

So if we doubled the sides of a triangle then area will increased by four times .