The area of a triangle is 84 cm². If the length of the sides are consecutive natural no.s, then what are the dimensions (in cm) of the triangle?
Answers
Step-by-step explanation:
Given :-
The area of a triangle is 84 cm². If the length of the sides are consecutive natural numbers.
To find:-
What are the dimensions (in cm) of the triangle?
Solution :-
Given that :
The lengths of the sides are consecutive natural numbers.
Let the lengths of the three sides are
(X-1) cm X cm ,(X+1) cm
Let a = (X-1) cm
Let b = X cm
Let c = (X+1) cm
We know that
Area of a triangle whose lengths of its sides are a units , b units and c units is by Heron's formula
∆ = √[S(S-a)(S-b)(S-c)] sq.units
Where S = (a+b+c)/2 units
Now,
S = (X-1+X+1+X)/2
=> S = 3X/2 cm
I) S-a = (3X/2) - (X-1)
=> S-a =(3X-2(X-1))/2
=> S-a = (3X-2X+2)/2
=> S-a =( X+2)/2 cm
ii) S-b = ((3X/2) -X)
=> S-b =(3X-2X)/2
=> S-b = X/2 cm
iii) S-c = (3X/2) - (X+1))
=> S-c = (3X-2(X+1))/2
=> S-c =(3X-2X-2)/2
=> S-c = (X-2)/2 cm
Now , Area of the given triangle
=> ∆ = √[(3X/2){(X+2)/2}{X/2}{(X-2)/2}]
=> ∆ = √[{(3X)(X+2)(X)(X-2)}/(2×2×2×2)]
=> ∆ = √[(3X)(X+2)(X)(X-2)/16] cm²
According to the given problem
The area of the triangle = 84 cm²
=> √[(3X)(X+2)(X)(X-2)/16]= 84
On squaring both sides then
=>(3X)(X)(X+2)(X-2)/16= 84²
=>(3/16) (X)(X)(X+2)(X-2) = 7056
=> X²(X+2)(X-2) = 7056×16/3
=> X²(X+2)(X-2) = 2352×16
=> X²(X+2)(X-2)= 37632
=> X²(X+2)(X-2)= 37632
=> X²(X²-4)= 37632
Since (a+b)(a-b) = a²-b²
=> X⁴-4X² = 37632
On adding 4 both sides then
=> X⁴-4X²+4 = 37632+4
=>(X²)²-2(X²)(2)+2² = 37636
=> (X²-2)² = 37636
Since (a-b)² = a²-2ab+b²
Where , a = X² and b = 2
=> (X²-2)² = ±√37636
=> (X²-2) = ±194
=> X² = ±194+2
=> X² = 194+2 or -194+2
=> X² = 196 or -192
=> X= ±√196 or ±√-192
=> X = ±14 ,X cannot be±√-192
X can not be negative
=> X = 14 cm
Now ,
X-1 = 14-1 = 13 cm
X+1 = 14+1 = 15 cm
The lengths of the sides = 13 cm ,14 cm and 15 cm
Answer:-
The dimensions of the given triangle are
13 cm ,14 cm and 15 cm
Check:-
The sides are 13 cm ,14 cm and 15 cm
S = (13+14+15)/2 = 42/2 = 21 cm
Area = √[S(S-a)(S-b)(S-c)] sq.units
=> ∆ = √[21(21-13)(21-14)(21-15)]
=>∆ =√(21×8×7×6)
=> ∆ = √(3×7×2×2×2×7×2×3)
=> ∆ =√[(2×2)×(2×2)×(3×3)×(7×7)]
=> ∆ = 2×2×3×7
=> ∆ = 84 cm²
Verified the given relations in the given problem
Used formulae:-
- Area of a triangle whose lengths of its sides are a units , b units and c units is by Heron's formula
- ∆ = √[S(S-a)(S-b)(S-c)] sq.units
- Where S = (a+b+c)/2 units
- (a-b)² = a²-2ab+b²
- (a+b)(a-b) = a²-b²