Question

The cost of turfing a triangular field at the rate of Rs. 5 per sq.m is Rs. 1350 .If the sides of the field are in the ratio of 5:12:13. Find the sides of field .​

Answer 1

Given :-

Cost of turfing a triangular field @ Rupees 5 per m² is rupees 1350.

Therefore area of the triangular field = 1350/5

= 270m²

Now,

Ratio between the sides of the triangular field is given 5 : 12 : 13

Let the sides of the triangular field be 5x, 12x and 13x respectively.

➡ Semi-perimeter of the triangle = (5x + 12x + 13x)/2

= 15x

By using heron's formula, we get

√s(s - a)(s - b)(s - c) = area of the triangle where s is the semi-perimeter and a, b and c are it's sides respectively.

➡ √[15x(15x - 5x)(15x - 12x)(15x - 13x)] = 270m²

➡ √(15x × 10x × 3x × 2x) = 270m²

➡ √(900x⁴) = 270m²

➡ 30x² = 270m²

➡ x² = 270/30

➡ x² = 9

➡ x = √9

➡ x = 3

Hence, the sides of the triangular field are :-

• 5x = 5 × 3 = 15m

• 12x = 12 × 3 = 36m

• 13x = 13 × 3 = 39m

Answer 2

Answer :

The sides of the triangular field are :

• 15m.
• 36m.
• 39m.

Step-by-step explanation :

Given :

Cost of turfing a triangular field Rs. 5 per meter² (m)² is Rs. 1350.

It implies,

Area of the triangular field = $\sf \dfrac{1350}{5}$

=> 270m².

Ratio between the sides of the triangular field - 5 : 12 : 13

Solution :

★ Consider the -

Sides of the triangular field as -  5y, 12y and 13y.

We have to find first - Semi-perimeter of the triangle.

So,

$\sf\\\implies \dfrac{(5y + 12y + 13y)}{2}\\ \\\implies 15y$

We know that :

★ Heron's Formula :

$\bigstar\;\boxed{\sf\sqrt {s(s - a)(s - b)(s - c)} }}$

Plug the given values :

$\sf\\\implies \sqrt{[15y(15y - 5y)(15y - 12y)(15y - 13y)]}= 270m^2\\ \\\implies \sqrt{(15y \times 10y \times 3y \times 2y)} = 270m^2\\ \\\implies \sqrt{(900y^4)} = 270m^2\\ \\\implies 30y^2 = 270m^2\\ \\\implies y^2 = \dfrac{270}{30}\\ \\\implies y^2 = 9\\ \\\implies y = \sqrt 9\\ \\\implies y = 3$

∴ The value of y - '3'

Now,

$\sf \\\implies 5y = 5 \times 3 = 15m\\ \\\implies 12y = 12 \times 3 = 36m\\ \\\implies 13y = 13 \times 3 = 39m$

The sides of the triangular field are :

• 15m
• 36m
• 39m