Question

A traffic signal board, including"school AHead;is an equilateral triangle with side "a". find the area of singnal board, using Heron, s ​

Answer 1

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..

Answer 2

${\bold{\blue{\underline{\red{Corr}\pink{ect}\:\green{Quest}\purple{ion:-}}}}}$

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 it's perimeter is 180 cm ,what will be the area of the signal board?

${\bold{\pink{\underline{\green{G}\purple{iv}\orange{en}\red{:-}}}}}$

• Perimeter of triangle = 180 cm

• side of equilateral triangle = a

${\bold{\blue{\underline{\red{To}\:\pink{Fin}\green{d}\purple{:-}}}}}$

• Area of the signal board = ?

${\bold{\pink{\underline{\red{So}\purple{lut}\green{ion}\orange{:-}}}}}$

$\bf\:semi\:perimeter=\dfrac{a+b+c}{2}$

$\bf\:semi\:perimeter=\dfrac{a+a+a}{2}$

$\bf\:semi\:perimeter=\dfrac{3a}{2}$

$\bf\boxed\star\purple{\underline{\underline{{Using\:Heron's\: formula:-}}}}$

$\bf\:Area\:of\triangle\:ABC=\sqrt{s(s-a)(s-b)(s-c)}$

$\bf\:Area\:of\triangle\:ABC=\sqrt{\dfrac{3a}{2}\times\:\dfrac{3a-2a}{2}\times\:\dfrac{3a-2a}{2}\times\:\dfrac{3a-2a}{2}}$

$\bf\:Area\:of\triangle\:ABC=\sqrt{\dfrac{3a}{2}\times\:\dfrac{a}{2}\times\:\dfrac{a}{2}\times\:\dfrac{a}{2}}$

$\bf\:Area\:of\triangle\:ABC=\dfrac{1}{2}\times\:\dfrac{1}{2}\times\:a\times\:a\sqrt{3}$

$\bf\:Area\:of\triangle\:ABC=\dfrac{1}{4}\times\:\sqrt{3}\times\:{a}^{2}$

$\bf\:Area\:of\triangle\:ABC=\dfrac{\sqrt{3}}{4}{a}^{2}$

Now,

$\bf\:perimeter\:of\:triangle=180\:cm$

$\bf\:side\:of\:triangle=\dfrac{180}{3}$

$\bf\:side\:of\:triangle=60\:cm$

Area of an equilateral triangle$\bf\:=\dfrac{\sqrt{3}}{4}{a}^{2}$

Area of an equilateral triangle = $\bf\:=\dfrac{\sqrt{3}}{4}\times\:({60})^{2}$

Area of an equilateral triangle = 900√3 cm²