Question

the perameter of a triangular feild is 432 cm and its sides are in the ratio 12:17:25 find its area​

Answer 1

Given :

• Perimeter of triangular field = 432cm
• Ratio of side = 12 : 17 : 25

To find :

Area of triangle

Formula used :

$\sf \bullet \: Perimeter \: of \: triangle \: = \: sum \: of \: all \: side \\ \\ \sf \bullet \: area \: of \: triangle \: = \: \sqrt{s(s - a)(s - b)(s - c)}$

Here,

• S = semi-perimeter
• a, b , c are side of triangle

SOLUTION :

Let side be a, b ,c

➝ a : b : c = 12 : 17 : 25

Let ,

➝ a = 12x

➝ b = 17x

➝ c = 25x

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➝ Perimeter of triangle = a + b + c

➝ Perimeter of triangle = 12x + 17x + 25x

➝ Perimeter of triangle = 54x

➝ 432cm = 54x

➝ x = 432cm ÷ 54

➝ x = 8cm

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• a = 12x

➝ a = 12×8cm

➝ a = 96 cm

• b = 17x

➝ b = 17 × 8cm

➝ b = 136 cm

• c = 25x

➝ c = 25 × 8 cm

➝ c = 200cm

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Now , semi-perimeter (s) = perimeter ÷ 2

➝ s = 432 ÷ 2

➝ s = 216cm

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

$\sf \implies \: Area \: of \: triangle \: = \sqrt{216 \times (216 - 96) \times (216 - 136) \times (216 - 200)}$

$\sf \implies \: Area \: of \: triangle \: = \: \sqrt{216 \: \times \: 120 \: \times 80 \: \times 16 \: }$

$\sf \implies \: Area \: of \: triangle \: = \sqrt{(36 \times 6) \times \: (6 \times 20) \times (20 \times 4) \times (16) }$

$\sf \implies \: Area \: of \: triangle \: = \sqrt{(6 \times 6) \times (6 \times \: 6) \times (20\times 20)\times( 2 \times 2) \times (4 \times 4) }$

$\sf \implies \: Area \: of \: triangle \: = (6 ) \times (6 ) \times ( 20)\times( 2 ) \times (4 )$

$\sf \implies \: Area \: of \: triangle \: = 5760$

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ANSWER : 5760 cm²

Answer 2

${\large {\underbrace{ \underline{ \text{Understanding \: the \: Question}}}}}$

Here's the concept of perimeter of triangle as well as area of triangle is used. We have given ratios of sides of triangle and perimeter,so we can easily find the sides of triangle. To find the area we will use“ Heron's formula”

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So let's start!

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GIVEN:

➛Perimeter of triangle field =432cm

➛Ratio of sides of triangle are 12:17:25

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TO FIND:

➛Area of triangle

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SOLUTION:

Let the ratios of sides be 12x:17x:25x.

${\mapsto{ \boxed{ \sf{Perimeter \: of \: triangle=Sum \: of \: all \: sides}}}}$

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$\sf{\implies 432cm=12x + 17x + 25x}$

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$\sf{\implies 432cm=54x}$

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$\sf{\implies 8cm = x}$

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So length of sides are:-

➢12x=12×8cm=96cm

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➢17x=17×8cm=136cm

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➢25x=25×8cm=200cm

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Let's name these sides

• A=96cm
• B=136cm
• C=200cm

Now we have to find semi-perimeter to apply heron's formula.

$\mapsto{ \boxed{\sf{S= \dfrac{A+B+C}{2}}}}$

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${\sf{\implies S= \dfrac{96cm + 136cm + 200cm}{2}}}$

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${\sf{\implies S= \dfrac{432cm}{2}}}$

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${\sf{\implies S= 216cm}}$

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$\large{{ \underline{ \underline{ \mathbb{HERON'S \: \: FORMULA}}}}}$

${ \mapsto{ \boxed{ \sf{Ar. \:triangle= \sqrt{S(S-A)(S-B)(S-C)}}}}}$

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${ \sf{Ar. \:triangle= \sqrt{216(216 - 96)(216-136)(216- 200)}}}$

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${ \sf{Ar. \:triangle= \sqrt{216(120)(80)(16)}}}$

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${ \sf{Ar. \:triangle= \sqrt{33,177,600}}}$

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Ar. triangle=5760cm².

∴The required area is 5760cm².

${\large{\underline{ \underline{ \large{ \text{More \:formulae : }}}}}}$

》Area of square=Side×Side

》Area of rectangle=Length×Breadth

》Area of ∆=½×Base×Height

》Perimeter of square=4×Side

》Perimeter of rectangle=2(L+B)

[Here L and B refers to length and breadth]