# If Every Side of a Triangle is Doubled, Then Increase in the Area of the Triangle is - Mathematics

MCQ

If every side of a triangle is doubled, then increase in the area of the triangle is

#### Options

• $100\sqrt{2}$ %

• 200%

•  300%

•  400%

#### Solution

The area of a triangle having sides aband s as semi-perimeter is given by,

A = sqrt(s(s-a)(s-b)(s-c)), where

s = (a+b+c)/2 ⇒ 2s = a + b + c

We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one

Now, the area of a triangle having sides 2a, 2b, and 2c and  s1 as semi-perimeter is given by,

A_1 = sqrt(s_1(s_1-2a)(s_1 - 2b)(s_1 - 2c)

Where,

s_1 = (2a +2b+2c)/2

s_1 = (2(a+b+c))/2

s= a + b + c

s= 2s

Now,

A_1 = sqrt(2s (2s-2a)(2s-2b)(2s-2c))

A
_1 = sqrt(2s xx 2 ( s-a)xx 2 ( s-b) xx 2 (s-c))

A_1 = 4 sqrt(s (s-a)(s-b)(s-c))

A_1 = 4A

Therefore, increase in the area of the triangle

=A1 - A

=4A - A

=3 A Percentage increase in area

= (3A)/A xx 100

= 300 %

Concept: Application of Heron’s Formula in Finding Areas of Quadrilaterals
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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 17 Heron’s Formula
Exercise 17.4 | Q 14 | Page 25

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