Question

Ncert Solutions for class 9 maths Herons Formula

ex1
q1

last 200 point ​

Answer 1

Step-by-step explanation:

☞︎︎︎ Given :

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Side of equilateral △ = a units

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⇒ a = b = c

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↣ Now,

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Formula

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$\large \rm \bigstar \: \: s = \dfrac{a + b + c}{2}$

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↣ Here,

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a = a

b = a

c = a

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$\large \rm \implies \: \: s = \dfrac{a +a + a}{2}$

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

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Area of triangle

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☆ Herons Formula

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$\large \boxed{\mathtt{ = \sqrt{s(s - a)(s - b)(s - c)} }}$

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$\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{3a}{2} - a\right)\left( \dfrac{3a}{2} - a\right)\left( \dfrac{3a}{2} - a\right)}$

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$\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{3a - 2a}{2} \right)\left( \dfrac{3a - 2a}{2} \right)\left( \dfrac{3a - 2a}{2} \right)}$

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$\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \left( \dfrac{a}{2} \right)\left( \dfrac{a}{2} \right)\left( \dfrac{ a}{2} \right)}$

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$\large \rm⇒ \: \sqrt{ \dfrac{3a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{ a}{2} }$

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$\large \rm⇒ \: \sqrt{ \dfrac{3{a }^{4} }{16} }$

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$\: \: \: \: \: \large \rm \therefore\: { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }$

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(derived Formula)

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$: \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }$

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➢ Perimeter of board = 180cm

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⇒ a + a + a = 180cm

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⇒ 3a = 180cm

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Transposing The Terms

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$\large \rm \: ⇒ \: a = \dfrac{180}{3}$

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$\large \rm \: ⇒ \: a = \dfrac{ \cancel{180} \: \: 60}{ \cancel3}$

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$\large \rm \therefore \: a = 60cm$

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$\: \: \: \: : \mapsto \: \: \: \large \rm area = { \dfrac{ \sqrt{ 3} \: {a }^{2} }{4} }$

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$\: \: : \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: {(60) }^{2} }{4} }$

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$\: \: : \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: \times 60 \times 60 }{4} }$

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$: \implies \large \rm area = { \dfrac{ \sqrt{ 3} \: \times 60 \times \cancel{60} 15}{ \cancel4} }$

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⇒ area = 900√3cm²

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Area = 901.73 cm²

Answer 2

Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.

Solution:

Let the sides of the triangle be

a = 12x cm, b = 17x cm, c = 25x cm

Perimeter of the triangle = 540 cm

Now, 12x + 17x + 25x = 540

⇒ 54x = 54 ⇒ x = 10

∴ a = (12 x10)cm = 120cm,

b = (17 x 10) cm = 170 cm

and c = (25 x 10)cm = 250 cm

Now, semi-perimeter, s = 5402cm = 270 cm

NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula Ex 12.1 Q5