Question

the sides of a triangle are in the ratio 3:5:7: and its perimeter is 150 metre find its area​

Answer 1

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Given:

Ratio of sides of triangle = 3:5:7

Perimeter = 150 meter

To Find:

Area of triangle = ?

Solution:

Let the sides of triangle be

a = 3x, b = 5x, c = 7x

Perimeter of triangle is equal to addition of all sides of triangle [a+b+c]

$\\ Perimeter \: \: = 3x + 5x + 7x \\ \\ 150 = 15x \\ \\ \dfrac{150}{15} = x \\ \\ 10 = x$

$\\ Substituting \: \: value \: \: of \: \: x \\ \\ a = 3x = 3 \times 10 = 30 \: m \\ \\ b = 5x = 5 \times 10 = 50 \: m \\ \\ c = 7x = 7 \times 10 = 70 \: m$

$\\ Using \: Herons \: Formula \\ \\ semiperimeter \: = \dfrac{perimeter}{2} \\ \\ = \dfrac{150}{2} \\ \\ = 75 \: cm$

$\\ Area \: of \: triangle \: = \\ \sqrt{s(s - a)(s - b)(s - c)} \\ \\ s \: \: is \: \: semiperimeter \\ \\ a,b \: \: and \: c \: \: are \: \: sides \: \: of \: triangle$

$\\ Area \: = \\ \sqrt{75(75 - 30)(75 - 50)(75 - 70)} \\ \\ = \sqrt{75(45)(25)(5)} \\ \\ = \sqrt{421875} \\ \\ = 375 \sqrt{3} \: m {}^{2}$

$\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 649.52 \: m {}^{2}$

Required answer:

Therefore, area of triangle is 375√3 m² or 649.52 m²