Question

(i) If the lengths of the sides of a triangle are in the ratio 3: 4:5 and its perimeter is
48 cm, find its area.
(ii) The sides of a triangular plot are in the ratio 3 : 5 : 7 and its perimeter is 300m
.Find its area. √3 = 1.732 .

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Answer 1

Step-by-step explanation:

first one is 96 and 2nd is 2598.08

Answer 2

$\huge{\underline{\rm{\orange{\bf{Solution \ I:-}}}}}$

$\large{\underline{\rm{\green{\bf{Given:-}}}}}$

The ratio of the length of the side of a triangle = 3:4:5

The perimeter of the triangle = 48 cm

$\large{\underline{\rm{\green{\bf{To \: Find:-}}}}}$

The area of the triangle.

$\large{\underline{\rm{\green{\bf{Answer:-}}}}}$

Given that,

Ratio = 3:4:5

Perimeter = 48 cm

Let us consider the sides of triangle to be 3x, 4x, 5x

Perimeter = $\sf 3x+4x+5x=48 \: cm$

Perimeter $\implies \sf 12x=48$

Finding the value of x,

$\implies \sf x=\dfrac{48}{12}$

$\implies \sf x=4$

Next, finding the sides of the triangle

$\longrightarrow \sf 3x=3 \times 4=12\: cm$

$\longrightarrow \sf 4x=4 \times 4=16 \: cm$

$\longrightarrow \sf 5x=5 \times 4=20 \: cm$

Semi perimeter,

$\sf \dfrac{Sum \ of \ sides \ of \ triangle}{2}$

Substituting these values,

$\sf \dfrac{12 +16+20}{2} =\dfrac{48}{2}$

$\sf =24 \: cm$

Using Heron's formula,

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

Substituting their values,

$\sf Area \ of \ triangle=\sqrt{24(24-12)(24-16)(24-20)}$

$\sf Area \ of \ triangle=\sqrt{ 24 \times 12 \times 8 \times 4}=\underline{\underline{9216 \: cm^{2}}}$

Therefore, the area of the triangle is 9216 cm²

$\huge{\underline{\rm{\orange{\bf{Solution \ II:-}}}}}$

$\large{\underline{\rm{\green{\bf{Given:-}}}}}$

The ratio of the length of the side of a triangle = 3:5:7

The perimeter of the triangle = 300 cm

$\large{\underline{\rm{\green{\bf{To \: Find:-}}}}}$

The area of the triangle.

$\large{\underline{\rm{\green{\bf{Answer:-}}}}}$

Given that,

Ratio = 3:5:7

Perimeter = 300 cm

Let us consider the sides of triangle to be 3x, 5x, 7x

Perimeter = $\sf 3x+5x+7x=300 \: cm$

Perimeter $\implies \sf 15x=300$

Finding the value of x,

$\implies \sf x=\dfrac{300}{15}$

$\implies \sf x=20$

Next, finding the sides of the triangle

$\longrightarrow \sf 3x=3 \times 20=60\: cm$

$\longrightarrow \sf 5x=5 \times 20=100 \: cm$

$\longrightarrow \sf 7x=7 \times 20=140 \: cm$

Semi perimeter = $\sf a+b+\dfrac{c}{2}$

Semi perimeter = $\implies \sf 60+100+\dfrac{140}{2}$

$\implies \sf \dfrac{300}{5}$

$\implies \sf 150$

Using Heron's formula,

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

Substituting their values,

$\sf Area \ of \ triangle=\sqrt{150(150-60)(150-100)(150-140)}$

$\sf Area \ of \ triangle=\sqrt{150\times 90\times 50\times 10}=\underline{\underline{1500\sqrt{3} }}$

Therefore, the area of the triangle is 1500 √3 cm²