NCERT Solutions for class 9 maths.
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Step-by-step explanation:
Given :
\:
Side of equilateral △ = a units
\:
⇒ a = b = c
\:
↣ Now,
\:
Formula
\:
\large \rm \bigstar \: \: s = \dfrac{a + b + c}{2}★s=
2
a+b+c
\:
↣ Here,
\:
a = a
b = a
c = a
\:
\large \rm \implies \: \: s = \dfrac{a +a + a}{2}⟹s=
2
a+a+a
\:
\large \rm \implies \: \: s = \dfrac{3a}{2}⟹s=
2
3a
\:
Area of triangle
\:
☆ Herons Formula
\:
\large \boxed{\mathtt{ = \sqrt{s(s - a)(s - b)(s - c)} }}
=
s(s−a)(s−b)(s−c)
\:
\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{3a}{2} - a\right)\left( \dfrac{3a}{2} - a\right)\left( \dfrac{3a}{2} - a\right)}⇒
2
3a
(
2
3a
−a)(
2
3a
−a)(
2
3a
−a)
\:
\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{3a - 2a}{2} \right)\left( \dfrac{3a - 2a}{2} \right)\left( \dfrac{3a - 2a}{2} \right)}⇒
2
3a
(
2
3a−2a
)(
2
3a−2a
)(
2
3a−2a
)
\:
\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{a}{2} \right)\left( \dfrac{a}{2} \right)\left( \dfrac{ a}{2} \right)}⇒
2
3a
(
2
a
)(
2
a
)(
2
a
)
\:
\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{ a}{2} }⇒
2
3a
×
2
a
×
2
a
×
2
a
\:
\large \rm⇒ \: \sqrt{ \dfrac{3{a }^{4} }{16} }⇒
16
3a
4
\:
\: \: \: \: \: \large \rm \therefore\: { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }∴
4
3
a
2
\:
(derived Formula)
\:
: \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }:⟹area=
4
3
a
2
\:
➢ Perimeter of board = 180cm
\:
⇒ a + a + a = 180cm
\:
⇒ 3a = 180cm
\:
Transposing The Terms
\:
\large \rm \: ⇒ \: a = \dfrac{180}{3}⇒a=
3
180
\:
\large \rm \: ⇒ \: a = \dfrac{ \cancel{180} \: \: 60}{ \cancel3}⇒a=
3
180
60
\:
\large \rm \therefore \: a = 60cm∴a=60cm
\:
\: \: \: \: : \mapsto \: \: \: \large \rm area = { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }:↦area=
4
3
a
2
\:
\: \: : \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: {(60) }^{2} }{4} }:⟹area=
4
3
(60)
2
\:
\: \: : \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: \times 60 \times 60 }{4} }:⟹area=
4
3
×60×60
\:
: \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: \times 60 \times \cancel{60} 15}{ \cancel4} }:⟹area=
4
3
×60×
60
15
\:
⇒ area = 900√3cm²
\:
Area = 901.73 cm²