Question

The area of a triangle is 84 cm². If the length of the sides are consecutive natural numbers, then what are the dimensions (in cm) of the triangle?
1. 5,6,7
2. 9,10,11
3. 13,14,15
4. 15,16,17​

Answer 1

let the length of sides are x , x+1 , x+2

perimeter = x + x+1 + x+2 = 3x+3

Area = √s(s-a)(s-b)(s-c)

= √3x+3(3x+3 -x)(3x+3-x-1)(3x+3 -x-2)

=√3x+3(2x+3)(2x+2)(2x+1)

given= 84cm²

√3x+3(2x+3)(2x+2)(2x+1)= 84cm²

= 3x+3(2x+3)(2x+2)(2x+1) = √84

Answer 2

Given,

Area of the triangle = 84 cm²

To find,

Dimensions of the triangle

Solution,

We may use the following mathematical procedure to solve the issue.

The following is the procedure for determining the triangle's dimensions.

Let the dimensions of the sides of the triangle be x-1, x, and x+1.

We know that,

∴ Semi perimeter = $\frac{a+b+c}{2}$

                          =  $\frac{3x}{2}$  

∴ Area of triangle =√(s.(s-a).(s-b).(s-c))

⇒ 84 = √ ($(\frac{3x}{2} . \frac{(x+2)}{2} \frac{x}{2} \frac{(x-2)}{2} )$

⇒ 84 = $\frac{1}{4}$ √${3x^2(x^2-4)}$

Squaring both sides.

⇒ 336 x 336 = 3x²(x²-4)

⇒ 112 x 336 = x²(x²-4)

⇒ 37632 = x²(x²-4)

⇒ 37632 = (x²-2)²

⇒ (x²-2) = ± 194

⇒ x² = 196 or -192

Now, -192 is not acceptable.

⇒ x = 14 cm

Thus,

As a result, the triangle's sides have lengths of x-1, x, and x+1, or 13 cm, 14 cm, and 15 cm, respectively. (Option 3)