The lengths of the sides of Δ *ABC* are consecutive integers. It Δ *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 Δ*ABC**?*

#### Options

4

6

8

10

#### Solution

We are given that triangle ABC has equal perimeter as to the perimeter of an equilateral triangle having side 9 cm. The sides of triangle ABC are consecutive integers. We are asked to find the smallest side of the triangle ABC

Perimeter of an equilateral triangle, say *P *having side 9 cm is given by

p = 3a

a = 9 cm

p = 3 × 9

p = 27 cm

Let us assume the three sides of triangle ABC be *x,* *x+*1, *x−*1

Perimeter of triangle ABC, say *P*_{1 }is given by

*P*_{1}_{ }*=* AB + BC + AC

AB = *x*; BC =* x* +1; AC = *x−*1. Since *P _{1}*

_{ }=

*P*. So

`p_1 = p`

`27 = x + ( x+1) + (x-1) `

`27 = 3x`

`x = 27/3 `

x = 9 cm

By using the value of *x,* we get the sides of triangle as 8 cm, 9 cm and 10 cm