Question

The sides of a triangle field are in the ratio 5:3:4 and it's perimeter is 180m. Find:
1) it's area.
2) altitude of the triangle corresponding to its largest side.
3) the cost of levelling the field at the rate of rupees 10 per square metre.​

Answer 1

Given :-

The ratio of the sides is 5 : 3 : 4

Perimeter of the field = 180 meters

Required to find :-

• Area ?

• Altitude of the triangle corresponding to the longest side ?

• The cost of levelling the field at the rate of rupees 10 per square meter ?

Formula used :-

Heron's Formula

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

$\tt{ Semi-Perimeter = \dfrac{perimeter}{2} }$

$\rule{200}{2}$

$\tt{ Area \; of \; a \; triangle = \dfrac{1}{2} \times base \times \; height }$

Solution :-

Given ratio of the sides :- 5 : 3 : 4

So,

Let the sides be , 5x , 3x & 4x ( where " x " is an integer )

So,

we know that :-

perimeter of the triangle = Sum of all its sides

180 = 5x + 3x + 4x

180 = 12x

12x = 180

x = 180/12

x = 15

Hence,

The sides are ;

• 5x = 5 ( 15 ) = 75 m

• 3x = 3 ( 15 ) = 45 m

• 4x = 4 ( 15 ) = 60 m

Using the formula ;

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

here,

s - semi perimeter

a , b , c - three sides of the triangle

So,

we need to find the semi-perimeter

using the formula,

$\tt{ Semi-Perimeter = \dfrac{perimeter}{2} }$

So,

Semi-Perimeter = 180/2

Semi-Perimeter = 90 meters

Substitute this values in the formula ;

$\tt{ Area \; of \; a \; triangle = \sqrt{ 90 ( 90 - 75 ) ( 90 - 45 ) ( 90 - 60 ) }}$

$\tt{ Area = \sqrt{ 90 ( 15 ) ( 45 ) ( 30 ) }}$

$\tt{ Area = \sqrt{ 15 \times 2 \times 3 \times 15 \times 15 \times 3 \times 15 \times 2 }}$

$\tt{ Area = \sqrt{ {3}^{2} \times {15}^{2} \times {2}^{2} \times {15}^{2} }}$

$\tt{ Area = 3 \times 15 \times 2 \times 15 }$

$\tt{ Area = 1, 350 {m}^{2}}$

Hence,

Area of the triangular field =

1, 350 m²

Similarly ,

We need to find the height corresponding to the longest side (base) .

So,

Longest side ( base ) = 75 meters

using the formula ,

$\tt{ Area \; of \; a \; triangle = \dfrac{1}{2} \times base \times \; height }$

So,

Area of the triangle = 1, 350 m²

This implies ,

1350 m² = 1/2 x 75 x h

1350 x 2 = 75 x h

2700 = 75h

75h = 2700

h = 2700/75

h = 36 meters

Hence,

The Altitude of the triangle corresponding to the longest side is 36 meters .

At last ,

It is given that ;

Cost of levelling the field for m² = Rs. 10

Cost of levelling the field of 1350m² = ?

This implies ,

1350 x 10

=> 13, 500 rupees

Cost of levelling the whole field =

Rs. 13, 500