In triangle ABC, the length of BC is less than twice the length of AB by 3 cm. The length of the AC exceeds the length of AB by 1 cm. The perimeter of the triangle is 34 cm. The length (in cm) of the smallest side of the triangle is:

Given,

In triangle ABC, the length of BC is less than twice the length of AB by 3 cm

$$=$$>  BC = 2AB - 3 ............(1)

The length of the AC exceeds the length of AB by 1 cm

$$=$$>  AC = AB + 1 .............(2)

Perimeter of triangle = 34 cm

$$=$$>  AB + BC + AC = 34

$$=$$>  AB + 2AB - 3 + AB + 1 = 34

$$=$$>  4AB - 2 = 34

$$=$$>  4AB = 36

$$=$$>  AB = 9 cm

From (1), BC = 2AB - 3 = 2(9) - 3 = 15 cm

From (2), AC = AB + 1 = 9 + 1 = 10 cm

$$\therefore\ $$The length of the smaller side of the triangle = AB = 9 cm

Hence, the correct answer is Option C