# 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?

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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