Question

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

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Answer 1

⚘Solution:-

The the 12:17:25 be

$\sf{a=12 k}$

$\sf{b=17 k}$

$\sf{c=25 k}$

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Perimeter= $\sf{a+b+c}$

$\sf{540=12k+17k+25k}$

$\sf{540=54k}$

→$\sf \pink{k=10}$

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$\sf{a=12×10=120}$

$\sf{b=17×10=170}$

$\sf{c=25×10=250}$

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∴s=$\sf \dfrac{540}{2}$

∴$\sf \pink{s=270 cm}$

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By using heron's formula:

 $\sf\sqrt{s(s - a)(s - b)(s - c)}$

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 $\sf\sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

= $\sf\sqrt{270×150×100×20}$

=$\sf\sqrt{9×3×10×3×5×10×100×4×5}$

=$\sf{3×3×5×2×100}$

→$\sf \pink{9000cm²}$

Answer 2

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✯Given:

Ratio of sides of the triangle and its perimeter.

By using Heron’s formula, we can calculate the area of a triangle.

Heron's formula for the area of a triangle is:

$\sf{Area = \sqrt{s(s - a)(s - b)(s - c)} }$

Where a, b, and c are the sides of the triangle, and s = Semi-perimeter = Half the perimeter of the triangle

Since the ratios of the sides of the triangle are given as 12:17:25

So, we can assume the length of the sides of the triangle as 12x cm, 17x cm, and 25x cm.

So the perimeter of the triangle will be

Perimeter = 12x + 17x + 25x

12x + 17x + 25x = 540 (given)

54x = 540

54x = 540x = 540/54

54x = 540x = 540/54x = 10 cm

Therefore, the sides of the triangle:

12x = 12 × 10 = 120 cm, 17x = 17 × 10 = 170 cm, 25x = 25 × 10 = 250 cm

a = 120cm, b = 170 cm, c = 250 cm

Semi-perimeter(s) = 540/2 = 270 cm

By using Heron’s formula,

$\sf{Area \: of \: a \: triangle = \sqrt {s(s - a)(s - b)(s - c)} }\\ = \sf{\sqrt{270(270 - 120)(270 - 170)(270 - 250)}} \\ = \sf{ \sqrt{270 × 150 × 100 × 20}}\\ =\sf{\sqrt{81000000}}\\ \sf \red{ = 9000 cm^{2}} \\ \sf \red{ Area \: of \: the \: triangle = 9000 cm^{2}.}$

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