Question

1. 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?​

Answer 1

Answer :

›»› The area of the equilateral triangle signal board will be 900√3 cm².

Given :

• A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'.
• The perimeter of an equilateral triangle is 180 cm.

To Find :

• The area of equilateral triangle signal board.

Solution :

As it is given that, the signal board is in the shape of equilateral triangle. So, all sides of the equilateral triangle are equal.

As we know that

→ Perimeter of traingle = Sum of all sides.

→ 180 = a + a + a

→ 180 = 2a + a

→ 180 = 3a

→ a = 180/3

→ a = 60

The each side of an equilateral triangle of is 60 cm.

As we know that

→ Semi perimeter = Perimeter ÷ 2

→ Semi perimeter = 180 ÷ 2

→ Semi perimeter = 90

The semi perimeter of an equilateral triangle is 90 cm.

Now,

As we know that

→ Area = √{s(s - a)(s - b)(s - c)}

→ Area = √{90(90 - 60)(90 - 60)(90 - 60)}

→ Area = √{90(30)(90 - 60)(90 - 60)}

→ Area = √{90(30)(30)(90 - 60)}

→ Area = √{90(30)(30)(30)}

→ Area = √{3 × 30 × 30 × 30 × 30}

→ Area = 30 × 30√3

→ Area = 900√3

Hence, the of the equilateral triangle signal board will be 900√3 cm².

Answer 2

Given :

• its perimeter is 180 cm,

To Find ;

• what will be the area of the signal board?

Solution :

Concept :

Heron's formula named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known.

• Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.

_____________________

Let 2s be the perimeter of the signal board. Then,

⠀⠀

$:\implies\sf 2s = a + a + a$

$:\implies\sf 2s = 3a$

$:\implies\sf s = \dfrac{3a}{2}$

⠀⠀⠀

Let A be the area of the given equilateral triangle, Then,

⠀⠀

Using Heron's Formula,

⠀⠀

• you can calculate the area of a triangle if you know the lengths of all three sides ,using a formula that has been known for nearly 2000 years it is called Heron's formula

$\;{\boxed{\sf{\sf{A = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$:\implies\sf A = \sqrt{ \dfrac{3a}{2} \bigg( \dfrac{3a}{2} - a \bigg) \bigg( \dfrac{3a}{2} - a \bigg) \bigg( \dfrac{3a}{2} - a \bigg)}\qquad\qquad\bigg\lgroup \because\;a = b = c\bigg\rgroup$

$:\implies\sf A = \sqrt{ \dfrac{3a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2}}$

$:\implies\sf A = \sqrt{ \dfrac{3a^4}{16}}$

$:\implies{\boxed{\sf{\sf{A = \dfrac{ \sqrt{3}}{4}a^2}}}}\;$

⠀⠀⠀

If, Perimeter = 180 cm. Then,

⠀⠀

$:\implies\sf 2s = 180$

$:\implies\sf \cancel{2} \times \dfrac{3a}{ \cancel{2}} = 180$

$:\implies\sf 3a = 180$

$:\implies{\boxed{\sf{\sf{a = 60}}}}\;$

Therefore,

⠀⠀

$:\implies\sf A = \dfrac{ \sqrt{3}}{4} \times (60)^2$

$:\implies\sf A = \dfrac{ \sqrt{3}}{ \cancel{4}} \times \cancel{3600}$

$:\implies{\sf{\sf{A = 900 \sqrt{3}\;cm^2}}}\;$

• Hence Area of signal board is 900√3 Cm²