Triangle ABC has sides $AB = 6.5$ cm, $BC = 7.8$ cm, and $AC = 9.1$ cm, measured to the nearest mm. (i) Express the length of side $AB$ as a range in the form $\le AB <$. (ii) Calculate the least possible perimeter of the triangle.

1. Problem Description

Triangle ABC has sides $AB = 6.5$ cm, $BC = 7.8$ cm, and $AC = 9.1$ cm, measured to the nearest mm.

(i) Express the length of side $AB$ as a range in the form $\le AB <$.

(ii) Calculate the least possible perimeter of the triangle.

2. Solution Steps

(i) The length $AB = 6.5$ cm is measured to the nearest mm, which is 0.1 cm.

Therefore, the lower bound is $6.5 - 0.05 = 6.45$ cm and the upper bound is $6.5 + 0.05 = 6.55$ cm.

So, $6.45 \le AB < 6.55$.

(ii) To calculate the least possible perimeter, we use the lower bounds for each side:

$AB_{min} = 6.45$ cm

$BC_{min} = 7.8 - 0.05 = 7.75$ cm

$AC_{min} = 9.1 - 0.05 = 9.05$ cm

The least possible perimeter is the sum of these minimum values:

$P_{min} = AB_{min} + BC_{min} + AC_{min} = 6.45 + 7.75 + 9.05 = 23.25$ cm.

3. Final Answer

(i) $6.45 \le AB < 6.55$

(ii) $23.25$ cm