Q. 214.4( 5 Votes )

The lengths of the sides of triangle ABC are consecutive integers. If triangle ABC has the same perimeter as an equilateral triangle with a side of length 9 cm, what is the length of the shortest side of triangle ABC?

Answer :

To Find: the shortest side of triangle ABC


Given: Length of the sides of a triangle are consecutive integers.


The perimeter of triangle = Perimeter of an equilateral triangle of side length 9 cm.


Concept Used:


If sides of a triangle are a, b, and c, then area of a triangle is given by:



Where s = semiperimeter of the triangle



Perimeter of equilateral triangle = 3 × side


Assumption:


Let the sides of the triangle be (x – 1), x, and (x + 1).


Explanation:


Perimeter of equilateral triangle = 3 × 9 cm = 27 cm


The perimeter of the given triangle = 27 cm


x – 1 + x + x + 1 = 27 cm


3 x = 27 cm


x = 9 cm


Therefore, the sides of the triangle are:


(x – 1) = (9 – 1) cm = 8 cm


x = 9 cm = 9 cm


(x + 1) cm = (9 + 1) cm = 10 cm


Now calculate the area of this triangle,








Hence, the area of the triangle is


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