Question

Chapter= Heron's formula
So the question is a traffic signal board indicating SCHOOL AHEAD is an equalateral triangle with side a' . Find the area of the signal board using Heron's formula .if it's perimeter is 180cm what will be the area of the signal board

Answer 1

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The perimeter of a triangle is equal to the sum of its three sides it is denoted by 2S.

2s=(a+b+c)

s=(a+b+c)/2

Here ,s is called semi perimeter of a triangle.

The formula given by Heron about the area of a triangle is known as Heron's formula. According to this formula area of a triangle= √s (s-a) (s-b) (s-c)

Where a, b and c are three sides of a triangle and s is a semi perimeter.

 

This formula can be used for any triangle to calculate its area and it is very useful when it is not possible to find the height of the triangle easily . Heron's formula is  generally used for calculating area of scalene triangle.

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 Solution :-

Given, side of a signal whose shape is an equilateral triangle= a

Semi perimeter, s=a+a+a/2= 3a/2

Using heron’s formula,

Area of the signal board = √s (s-a) (s-b) (s-c)

= √(3a/2) (3a/2 – a) (3a/2 – a) (3a/2 – a)

= √3a/2 × a/2 × a/2 × a/2

= √3a⁴/16

= √3a²/4

Hence, area of signal board with side a by using Herons formula is= √3a²/4

 

Now,

Perimeter of an equilateral triangle= 3a

Perimeter of the traffic signal board = 180 cm  (given)

 3a = 180 cm

 a = 180/3= 60 cm

Now , area of signal board=√3a²/4

= √3/4 × 60 × 60

= 900√3 cm²

Hence , the area of the signal board when perimeter is 180 cm is 900√3 cm².

 =========================================================

Hope this will help you....

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Answer 2

Long method :

The perimeter of a triangle is equal to the sum of its three sides it is denoted by 2S.

2s=(a+b+c)

s=(a+b+c)/2

Here ,s is called semi perimeter of a triangle.

The formula given by Heron about the area of a triangle is known as Heron's formula. According to this formula area of a triangle= √s (s-a) (s-b) (s-c)

Where a, b and c are three sides of a triangle and s is a semi perimeter.

This formula can be used for any triangle to calculate its area and it is very useful when it is not possible to find the height of the triangle easily . Heron's formula is  generally used for calculating area of scalene triangle.

____________________________________________________________

Solution:

Given, side of a signal whose shape is an equilateral triangle= a

Semi perimeter, s=a+a+a/2= 3a/2

Using heron’s formula,

Area of the signal board = √s (s-a) (s-b) (s-c)

= √(3a/2) (3a/2 – a) (3a/2 – a) (3a/2 – a)

= √3a/2 × a/2 × a/2 × a/2

= √3a⁴/16

= √3a²/4

Hence, area of signal board with side a by using Herons formula is= √3a²/4

Now,

Perimeter of an equilateral triangle= 3a

Perimeter of the traffic signal board = 180 cm  (given)

 3a = 180 cm

 a = 180/3= 60 cm

Now , area of signal board=√3a²/4

= √3/4 × 60 × 60

= 900√3 cm²

Hence , the area of the signal board when perimeter is 180 cm is 900√3 cm².

 =========================================================

Hope this will help you....

Short method :

Answer:

900√3

Step-by-step explanation:

Perimeter =180 cm

S=180/2=90 cm

Side=180/3=60 cm

Area=√S(s-a)(s-b)(s-c)

=√90(90-60)(90-60)(90-60)

=√90*30*30*30

=√30*3*30*30*30

=30*30√3

=900√3

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