Question

The perimeter of a triangular field is 540m and its sides are in the
ratio 5:12:13 .find the area of thetriangle​

Answer 1

Given sides of a triangular plot are in ratio 5:12:13.

Perimeter of plot is 540 m.

We have to find, Area of triangular plot.

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Let the sides of a triangular plot be 5x, 12x, and 13x.

According to the question,

we know that,

Perimeter of triangle = sum of measure's of it's all sides.

$:\implies\sf 12x+5x+13x=540$

$:\implies\sf 30x = 540$

$:\implies\sf x = \cancel{ \dfrac{540}{30}}$

$:\implies\sf x= 5$

Therefore, 

Measure of sides of triangle is,

5x = 5 × 20 = 25 m

12x = 12 × 20 = 60 m

13x = 13 × 20 = 65m

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✇ Using Heron's Formula,

$\star\;{\boxed{\sf{\pink{ \sqrt{s(s - a)(s - b)(s - c)}}}}}$

Where,

$\dashrightarrow\sf s = semi - perimeter$

$\dashrightarrow\sf s = \dfrac{a+b+c}{2}$

$\dashrightarrow\sf s = \cancel{ \dfrac{25+60+65}{2}}$

$\dashrightarrow\bf s = 75\;m$

Therefore,

$\sf Here \begin{cases} & \sf{a = 25\;m} \\ & \sf{b = 60\;m} \\ & \sf{c = 65\;m} \\ & \sf{s = 270\;m} \end{cases}$

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✇ Now, Putting Values in Formula,

$:\implies\sf \sqrt{75(75-25)(75-60)(75-65)}$

$:\implies\sf \sqrt{75 \times 50 \times 15 \times 10}$

$:\implies\sf \sqrt{750}$

$:\implies{\boxed{\frak{\purple{ \sqrt{750}\;m^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Thus,\;Area\;of\;given\; triangle\;is\; \bf{ \sqrt{750}\;m^2}.}}}$

Answer 2

$\huge\underline{\bf{Given}}$

• The perimeter of a triangular field is 540m.
• Its sides are in the ratio 5:12:13.

⠀

$\huge\underline{\bf{To\: Find}}$

• The area of triangle.

⠀

$\huge\underline{\bf{Concept}}$

• In this question, we will find the area of triangle by using heron's formula.
• For this we have to find the sides of the triangle.

⠀

$\huge\underline{\bf{Solution}}$

• Let the ratio be x.

⠀

★ Then, the sides will be

$\tt\longrightarrow{a = 5x}$

$\tt\longrightarrow{b = 12x}$

$\tt\longrightarrow{c = 13x}$

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★ We know that

⠀

$\small{\boxed{\tt{\bigstar{Perimeter\: of\: triangle = Sum\: of\: all\: the\: measures\: of\: its\: sides{\bigstar}}}}}$

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$\tt\longmapsto{Perimeter = 540}$

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$\tt\longmapsto{5x + 12x + 13x = 540}$

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$\tt\longmapsto{30x = 540}$

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$\tt\longmapsto{x = \dfrac{540}{30}}$

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$\tt\longmapsto{x = 18}$

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★ Therefore, measures of three sides are

→ a = 5 × 18 = 90m

→ b = 12 × 18 = 216m

→ c = 13 × 18 = 234m

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★ Finding semi - perimeter (s)

⠀

$\boxed{\tt{\bigstar{Semi - perimeter = \dfrac{a + b + c}{2}{\bigstar}}}}$

⠀

$\sf\implies{s = \dfrac{90 + 216 + 234}{2}}$

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$\sf\implies{s = \cancel{\dfrac{540}{2}}}$

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$\sf\implies{s = 270}$

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★ Heron's Formula

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$\boxed{\tt{\bigstar{Area = \sqrt{s(s - a) (s - b) (s - c)}{\bigstar}}}}$

⠀

Here,

• s = 270
• a = 90
• b = 216
• c = 234

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★ Putting the values

⠀

$\small\tt:\implies\: \: \: \: \: \: \: \: {Area = \sqrt{270(270 - 90) (270 - 216) (270 - 234)}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = \sqrt{270(180) (54) (36)}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = \sqrt{48,600 \times 54 \times 36}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = \sqrt{48,600 \times 1,944}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = \sqrt{94,478,400}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {Area = 9,720}$

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Hence, the area of the triangular field is 9,720 m².

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