Question

The sides of a triangular plot are in the ratio of 6 : 7 : 8 and its perimeter is 420 m. Find its area.

Answer 1

Given :

• Ratio of the sides of a triangular plot = 6 : 7 : 8
• Perimeter of the triangular plot = 420 m

To find :

• Area of the triangular plot

Knowledge required :-

• Formula of perimeter of triangle :-

⠀⠀⠀Perimeter = sum of all the sides

• Formula to calculate semi - perimeter (s) :-

⠀⠀⠀Semi - perimeter (s) = (a + b + c)/2

• Formula of area of triangle :-

⠀⠀⠀ Heron's formula = √s(s - a)(s - b)(s - c)

Solution :

Let the sides of the triangular plot be 6x, 7x and 8x.

• First side = 6x
• Second side = 7x
• Third side = 8x

⠀⠀⠀⇒ Perimeter = sum of all sides

⠀⠀⠀⇒ 420 = 6x + 7x + 8x

⠀⠀⠀⇒ 420 = 21x

⠀⠀⠀⇒ 420/21 = x

⠀⠀⠀⇒ 20 = x

The value of x = 20

Substitute the value of x in the sides which we have let.

• First side = 6x = 6 × 20 = 120 m
• Second side = 7x = 7 × 20 = 140 m
• Third side = 8x = 8 × 20 = 160 m

⠀⠀⠀⇒ Semi - perimeter (s) = (a + b + c)/2

Take,

• a = 120 m
• b = 140 m
• c = 160 m

⠀⠀⠀⇒ s = (120 + 140 + 160)/2

⠀⠀⠀⇒ s = 420/2

⠀⠀⠀⇒ s = 210

Semi - perimeter (s) = 210 m

⠀⇒ Heron's formula = √s(s - a)(s - b)(s - c)

⠀⠀⠀⇒ √210(210 - 120)(210 - 140)(210 - 160)

⠀⠀⠀⇒ √210(90)(70)(50)

⠀⠀⠀⇒ √(70 × 3 × 3 × 3 × 10 × 70 × 5 × 10)

⠀⠀⠀⇒ 70 × 3 × 10√(3 × 5)

⠀⠀⠀⇒ 2100√150

⠀⠀⠀⇒ 2100√(3 × 5)

⠀⠀⠀⇒ 2100√15

⠀⠀⠀⇒ The value of √15 = 3.88

⠀⠀⠀⇒ 2100 × 3.88

⠀⠀⠀⇒ 8148

★ Area of the triangular plot = 8148 m²

Answer 2

Step-by-step explanation:

Question:-

The sides of a triangular plot are in ratio 6:7:8 and it's perimeter is 420m. Find its area.

Answer:-

Given :-

• The given figure is a triangle.

• The ratio of its sides is 6:7:8.

• The perimeter of the triangle is 420m.

To find :-

Area of the triangle

Process :-

Let the sides of the triangle be 6x,7x and 8x.

∴ We know that :-

$⇒ Perimeter = \frac{a+b+c}{2}$

Inserting the above sum in the formula :-

$⇒ Perimeter = \frac{a+b+c}{2} \\ \\ ⇒420 = \frac{6x +7x + 8x}{2} \\ \\ ⇒ 420=\frac{21x}{2} \\ \\ ⇒ \frac{21x}{2} = 420 \\ \\ ⇒21x = 420 × 2 \\ \\ ⇒21x = 840 \\ \\ ⇒x = \frac{840}{21} \\ \\ ⇒x = 40$

Inserting the value of x:-

= 6x = 6 × 40 = 240m

= 7x = 7 × 40 = 280m

= 8x = 8 × 40 = 320m

∴ Area of the triangle using heron's formula :-

$⇒ \sqrt{s(s-a)(s-b)(s-c)} \\ \\ ⇒ \sqrt{420(420-240)(420-280)(420-320)} \\ \\ ⇒ \sqrt{420 × 180 \times 140 \times 100} \\ \\ ⇒ \sqrt{ 5 \times 4 \times 7 \times 3 \times 5 \times 4 \times3 \times 3 \times 5 \times 4 \times 7 \times 5 \times 4\times 5} \\ \\ ⇒354.96 {m}^{2} \: \: \: (Ans)$

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