Question

what will be the area of the triangle if it is doubled?

Answer 1

Then the area of the triangle is increased by a factor of 4 if the sides are doubled. The area of a triangle is given by (1/2)*base*height. If the length of the sides becomes double so does the height. As base and height are becoming double the new area is 4 times the original area.


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Answer 2

Let a,b,c be the sides of the original ∆ & s be its semi perimeter.

S= (a+b+c)/2

2s= a+b+c.................(1)

The sides of a new ∆ are 2a,2b,2c

[ given: Side is doubled]

Let s' be the new semi perimeter.

s'= (2a+2b+2c)/2

s'= 2(a+b+c) /2

s'= a+b+c

S'= 2s. ( From eq 1)......(2)


Let ∆= area of original triangle

∆= √s(s-a)(s-b)(s-c).........(3)


&

∆'= area of new Triangle

∆' = √s'(s'-2a)(s'-2b)(s'-2c)

∆'= √ 2s(2s-2a)(2s-2b)(2s-2c)

[From eq. 2]


∆'= √ 2s×2(s-a)×2(s-b)×2(s-c)

= √16s(s-a)(s-b)(s-c)

∆'= 4 √s(s-a)(s-b)(s-c)

∆'= 4∆. (From eq (3))

Increase in the area of the triangle= ∆'- ∆= 4∆ - 1∆= 3∆

%increase in area= (increase in the area of the triangle/ original area of the triangle)× 100

% increase in area= (3∆/∆)×100


% increase in area= 3×100=300 %


Hence, the percentage increase in the area of a triangle is 300%