The perimeter of a triangular field is 144 m and the ratio of the sides is 3 : 4 : 5. Find the area of the field.
Answers
Given : The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm.
Let the sides be a = 3x , b = 4x and c = 5x .
Perimeter of ∆ = a + b + c
⇒ 144 = 3x + 4x + 5x
⇒ 12x = 144
⇒ x = 144/12
⇒ x = 12
So , the Sides of a triangle are :
a = 3x = 3 × 12 = 36 m
b = 4x = 4 × 12 = 48 m
c = 5x = 5 × 12 = 60 m
Semi Perimeter of the ∆,s = (a + b + c) /2
Semi-perimeter (s) = (36 + 48 + 60)/2
s = 144/2
s = 72 m
Using Heron’s formula :
Area of the ∆ , A = √s (s - a) (s - b) (s - c)
A = √72(72 - 36)(72 - 48)(72 - 60)
A = √72 × (36) × (24) × (12)
A = √(36 × 2) (36) (12 × 2) × 12
A = √(36 × 36 × 12 × 12) × (2 × 2)
A = 36 × 12 × 2
A = 72 × 12
A = 864 cm²
Hence, the area of the field is 864 cm².
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Given : The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm.
Let the sides be a = 3x , b = 4x and c = 5x .
Perimeter of ∆ = a + b + c
⇒ 144 = 3x + 4x + 5x
⇒ 12x = 144
⇒ x = 144/12
⇒ x = 12
So , the Sides of a triangle are :
a = 3x = 3 × 12 = 36 m
b = 4x = 4 × 12 = 48 m
c = 5x = 5 × 12 = 60 m
Semi Perimeter of the ∆,s = (a + b + c) /2
Semi-perimeter (s) = (36 + 48 + 60)/2
s = 144/2
s = 72 m
Using Heron’s formula :
Area of the ∆ , A = √s (s - a) (s - b) (s - c)
A = √72(72 - 36)(72 - 48)(72 - 60)
A = √72 × (36) × (24) × (12)
A = √(36 × 2) (36) (12 × 2) × 12
A = √(36 × 36 × 12 × 12) × (2 × 2)
A = 36 × 12 × 2
A = 72 × 12
A = 864 cm²
Hence, the area of the field is 864 cm².
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