Question

$\sf question : -$


A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula.If its perimeter is 180 cm, what will be the area of the signal board?

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Answer 1

Answer:

Given Perimeter =180 cm

Semi-perimeter s=

2

180

s=90 cm

Since all sides of equilateral triangle are equal

So,

Perimeter =180

a+a+a=180

3a=180

a=60 cm

Area of triangle =

s(s−a)(s−b)(s−c)

=

90(90−60)(90−60)(90−60)

=

90×30×30×30

=

(9×3×3×3)×(10)

4

=

(9

2

)×3×(10)

4

=9×

3

×(10)

2

=9×100×

3

=900

3

Answer 2

Given :

• Perimeter Of Traingle = 180 cm
• Side of Traingle is a

Using Formula

• Heron's Formula = $\sf \sqrt{s(s-a) (s-b) (s-c)}$

To Find :

• Area Of Signal Board

Answer :

Let Each side of traffic signal board (equilateral Traingle ) = Sum Of All Sides = a + a + a = 3a

Now, For S,

We have to find semi-perimeter

$\longrightarrow$ Semi-Perimeter S = $\sf \frac{( a + b + c ) }{2}$

Where, a , b and c all are Sides Hence,

$\longrightarrow$ S = $\sf \frac{(a + a + a )}{2}$

$\longrightarrow$ S = $\sf \frac{3a}{2}$

Now, We already Know

Heron's Formula

• Area of triangle = $\sf \sqrt{S(S-a) (S-b) (S-c)}$

• Area of triangle = $\sf \sqrt{s(S-a)(S-a)(S-a)}$

• Area of triangle = $\sf \sqrt{\frac{3a}{2}\bigg( \frac{3a}{2} - a\bigg)\bigg( \frac{3a}{2} - a\bigg) \bigg( \frac{3a}{2} - a\bigg)}$

• Area of triangle = $\sf \sqrt{\frac{3a}{2}\bigg( \frac{3a}{2} - a\bigg)\bigg( \frac{3a}{2} - a\bigg) \bigg( \frac{3a}{2} - a\bigg)}$

• Area of triangle = $\sf \sqrt{\frac{3a}{2}\bigg( \frac{3a - 2a}{2} \bigg) \bigg( \frac{3a - 2a}{2} \bigg) \bigg( \frac{3a - 2a}{2} \bigg) }$

• Area of triangle = $\sf \sqrt{\frac{3a}{2}\bigg( \frac{a}{2} \bigg) \bigg( \frac{a}{2} \bigg) \bigg( \frac{a}{2} \bigg) }$

• Area of triangle = $\sf \sqrt{\frac{3a}{2}\bigg( \frac{a}{2} \bigg) \bigg( \frac{a}{2} \bigg) ^{2} }$

• Area of triangle = $\sf \bigg( \frac{a}{2} \bigg) \sqrt{\frac{3a}{2}\bigg( \frac{a}{2} \bigg) }$

• Area of triangle = $\sf \bigg( \frac{a}{2} \bigg) \sqrt{\frac{3a^{2} }{2 ^{2} } }$

• Area of triangle = $\sf \bigg( \frac{a}{2} \bigg)\frac{ \sqrt{3} a }{2 }$

• Area of triangle = $\sf \underline{ \frac{ \sqrt{3} a ^{2} }{4 } }$

Hence,Area of the Signal Board = $\sf \frac{ \sqrt{3} a ^{2} }{4 }$

Now, Given Perimeter = 180 cm

For, Each side a = $\frac{180 }{3}$

a = $\frac{\cancel{180}^{\: \: 60×\cancel{3}} }{\cancel3}$

a = 60 cm

Now, Finally For, Area of the Signal Board = $\sf \frac{ \sqrt{3} a ^{2} }{4 }$

$\longrightarrow$ Area of the Signal Board = $\sf \frac{ \sqrt{3} (60) ^{2} }{4 }$

$\longrightarrow$ Area of the Signal Board = $\sf \frac{ \sqrt{3} (3600) }{4 }$

$\longrightarrow$ Area of the Signal Board = $\sf \frac{ \sqrt{3} \cancel{(3600)}^{\: \: 900×\cancel4} }{\cancel{4 }}$

$\longrightarrow$ Area of the Signal Board = $\underline{\boxed{ \bf 900\sqrt{3}}}$

Hence,

The Area of the Signal Board = $\sf 900\sqrt{3}$

I hope it helps you...