Find the percentage increase in the area of a triangle if its each side is doubled?

Question

Answer

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)..... And

∆' = 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%...

Answer

Percentage Increase in Area of a Triangle When its Each Side is Doubled

Explanation:

Let's assume that we have a triangle with sides a, b, and c. Its area can be calculated using Heron's formula:

A = √(s(s-a)(s-b)(s-c))

where s = (a+b+c)/2 is the semi-perimeter of the triangle.

Now, if we double each side of the triangle to get sides 2a, 2b, and 2c, the new semi-perimeter will be:

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

Therefore, the new area of the triangle can be calculated using Heron's formula as:

A' = √(s'(s'-2a)(s'-2b)(s'-2c))

Expanding the terms inside the square root:

A' = √(s(s-a)(s-b)(s-c)) × √((s+a)(s+b-c)(s+c-b)(s+a-b))

Dividing A' by A:

A'_/A = √((s+a)(s+b-c)(s+c-b)(s+a-b))/√(s(s-a)(s-b)(s-c))

Now, using the fact that s = (a+b+c)/2 and s' = a+b+c, we can simplify the expression as:

A'_/A = √(2a/2a) × √(2b/2b) × √(2c/2c) × √((a+b+c)(a+b-c)(a-b+c)(-a+b+c))/√((a+b+c)(-a+b+c)(a-b+c)(a+b-c))

Cancelling out the common terms:

A'_/A = √2 × √((a+b+c)(a+b-c)(a-b+c)(-a+b+c))/√((a+b+c)(-a+b+c)(a-b+c)(a+b-c))

A'_/A = √2 × √((a+b-c)(a-b+c)(b+c-a)(a+b+c))/√((a+b-c)(a-b+c)(b+c-a)(a+b+c))

A'_/A = √2

Conclusion:

As we can see, the area of the triangle is doubled when each of its sides is doubled. Therefore, the percentage increase in the area of the triangle is:

Percentage Increase = (A'_/A - 1) × 100%

Percentage Increase = (√2 - 1) × 100%

Percentage Increase ≈ 41.4%

Answer

300

Similar Class 9 Doubts

Find the area of equilateral triangle whose each sides is 'a' units by Heron's formula. also find the percantage increase in the area of triangle if its each side is doubled.

Find the area of triangle if its each side is doubled in percentage?

if each side of triangle is doubled the find the ratio of the areas of new triangle formed and the given triangle

If each side of an equilateral triangle is increased by 2 cm, then its area increases by 3√3 cm2. The length of its each side and its area are respectively equal to?

Find the percentage decrease in the area of the triangle if sides reduced to half?

Top Courses for Class 9View all

Science Class 9

Social Studies (SST) Class 9

Mathematics (Maths) Class 9

English Grammar Advanced

English Class 9

Top Courses for Class 9

Top Courses for Class 9

Suggested Free Tests

Related Content

Class 9 Study Material

Class 9 Subjects

Competitive Examinations for Class 9

NCERT Books

NCERT Solutions

Reference Books for Class 9

Other Popular Exams

Company