How many distinct rectangles with positive integer side lengths (in centimetres) can be drawn having a perimeter of 36 cm?

Introduction / Context:
This question involves counting the number of distinct rectangles with a fixed perimeter and integer side lengths. A rectangle is determined by its length and breadth, and a given perimeter imposes a linear relationship between these two side lengths. We must find all possible integer pairs that satisfy this relationship, while treating rectangles with interchanged side lengths as the same shape (since rotating a rectangle does not produce a new rectangle).

Given Data / Assumptions:
- Let the side lengths of the rectangle be a and b (in cm), both positive integers.
- The perimeter P is given by P = 2(a + b) and equals 36 cm.
- Therefore a + b = 18.
- Rectangles with sides (a, b) and (b, a) are considered the same rectangle because the shape is unchanged by swapping length and breadth.

Concept / Approach:
We need to find positive integer solutions to the equation a + b = 18. There are many ordered pairs (a, b), but we must identify distinct rectangles up to swapping sides. To do this, we can list all ordered solutions with a <= b, which ensures each rectangle is counted exactly once. The number of such pairs is the answer. Alternatively, we can note that as a runs from 1 up to 9 (where a and b meet or cross), b is determined as 18 - a, giving each distinct rectangle.

Step-by-Step Solution:
Step 1: Start with a + b = 18, where a and b are positive integers. Step 2: To avoid double-counting, impose a <= b. Step 3: List possible values of a and the corresponding b values:
a = 1 ⇒ b = 17,
a = 2 ⇒ b = 16,
a = 3 ⇒ b = 15,
a = 4 ⇒ b = 14,
a = 5 ⇒ b = 13,
a = 6 ⇒ b = 12,
a = 7 ⇒ b = 11,
a = 8 ⇒ b = 10,
a = 9 ⇒ b = 9. Step 4: For a > 9, we would get a > b, which we have already counted by symmetry when a < b. Step 5: This gives us 9 distinct pairs (a, b), hence 9 distinct rectangles.

Verification / Alternative check:
We can double-check by counting all ordered positive integer solutions to a + b = 18, which are 17 pairs (since a can be 1 through 17). Because each unordered pair (a, b) with a ≠ b appears twice in this list (once as (a, b) and once as (b, a)), and the pair (9, 9) appears only once, we can count: there are 8 pairs with unequal sides that correspond to 8 rectangles, plus one square with sides (9, 9). Thus 8 + 1 = 9 rectangles, matching our earlier list.

Why Other Options Are Wrong:
Options 8, 10, 11: These counts either undercount or overcount the actual number of distinct integer side pairs that satisfy the perimeter condition. Careful enumeration shows exactly 9 possibilities.
Option None of these: Incorrect because 9 is a listed option and has been established as the correct count.

Common Pitfalls:
A common mistake is to count ordered pairs without adjusting for symmetry, leading to an answer of 17 instead of 9. Another error is to stop listing too early and miss some valid pairs. Enforcing a <= b and systematically listing all possibilities ensures that each distinct rectangle is counted exactly once and no rectangles are missed or double-counted.

Final Answer:
There are 9 distinct rectangles with integer side lengths and perimeter 36 cm.