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

Solution:

Given, each side of a triangle is doubled

We have to find the area of the new triangle thus formed and the given triangle.

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

Area of triangle, A = √s(s - a)(s - b)(s - c)

Where s= semiperimeter

s = (a + b + c)/2 -------------------------- (1)

Now, the sides are doubled.

So, the new sides are 2a, 2b and 2c.

Semiperimeter, s = (2a + 2b + 2c)/2

= 2(a + b + c)/2

s = a + b + c

From (1),

New semiperimeter = 2s

Area of new triangle = √2s(2s - 2a)(2s - 2b)(2s - 2c)

= √2s[2(s - a)2(s - b)2(s - c)]

= √2s[8(s - a)(s - b)(s - c)]

= √16(s(s - a)(s - b)(s - c))

= 4√s(s - a)(s - b)(s - c)

= 4A

New area = 4A

Ratio of new area to the old area = new area/old area

= 4A/A

= 4/1

Therefore, the required ratio is 4:1

✦ Try This: If each side of a triangle is reduced to half, then find the ratio of area of the new triangle thus formed and the given triangle.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 12

NCERT Exemplar Class 9 Maths Exercise 12.4 Sample Problem 1

Summary:

If each side of a triangle is doubled, then the ratio of area of the new triangle thus formed and the given triangle is 4:1

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