Question

The sides of a triangular park are in the ratio of 2: 6: 7 and its perimeter is

300 m. Then its area is:

(A) 154 √57 cm^2

(B) 215 √45 cm^2

(C)
340 √56 cm^2

(D) 300√55cm^2​

Answer 1

• The area of a triangular park = 300√55 m².

Given :

• The ratio of the sides of a triangular park = 2 : 6 : 7.
• The perimeter of a triangular park = 300 m.

To Find :

• The area of a triangular park.

Solution :

First, we need to find the sides of a triangular park.

Let,

The first side be 2x.

The second side be 6x.

The third side be 7x.

Given,

Perimeter of the park = 300 m

→ a + b + c = 300 m

Where, a, b and c are the sides of a triangular park.

So,

→ 2x + 6x + 7x = 300 m

→ 8x + 7x = 300 m

→ 15x = 300 m

→ x = $\sf \cancel\dfrac{300}{15} \: m$

→ x = 20 m

So, the sides of a triangular park :

The first side = 2x = 2 × 20 m = 40 m.

The second side = 6x = 6 × 20 m = 120 m.

The third side = 7x = 7 × 20 m = 140 m.

Now, we have to find the area of a triangular park.

By heron's formula,

→ $\sqrt{s(s - a)(s - b)(s - c)}$

Where,

→ $s = \dfrac{a + b + c}{2}$

We have,

• a = 40 m.
• b = 120 m.
• c = 140 m.

→ $\dfrac{40 + 120 + 140}{2} \: m$

→ $\dfrac{160 + 140}{2} \: m$

→ $\cancel\dfrac{300}{2} \: m$

→ $150 \: m$

Now we have,

• a = 40 m.
• b = 120 m.
• c = 140 m.
• s = 150 m.

Now, substitute all the values in the heron's formula.

→ $\sqrt{150 \: m(150 \: m - 40 \: m)(150 \: m - 120 \: m)(150 \: m - 140 \: m)}$

→ $\sqrt{150(110)(30)(10) \: {m}^{4} }$

→ $\sqrt{(2 \times 3 \times 5 \times 5)(2 \times 5 \times 11)(2 \times 3 \times 5)(2 \times 5)} \: {m}^{2}$

→ $\sqrt{2 \times 3 \times 5 \times 5 \times 2 \times 5 \times 11 \times 2 \times 3 \times 5 \times 2 \times 5} \: {m}^{2}$

→ $\sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5 \times 11} \: {m}^{2}$

→ $2 \times 2 \times 3 \times 5 \times 5 \sqrt{5 \times 11} \: {m}^{2}$

→ $4 \times 15 \times 5 \sqrt{55} \: {m}^{2}$

→ $60 \times 5 \sqrt{55} \: {m}^{2}$

→ $300 \sqrt{55} \: {m}^{2}$

Hence,

The area of a triangular park is 300√55 m².

So, the option (D) 300√55 m² is correct.