Coordinate Geometry

Given: A table lamp is placed on a study table.

Concept Used: To describe the position of any object in a plane, we need two reference lines (perpendicular to each other) and the distances of the object from those lines.

Solution:

Step 1: Consider the study table as a plane surface.

Step 2: Choose any two adjacent edges of the table as reference lines. Let one edge along the length of the table be the reference line in the horizontal direction, and one edge along the width be the reference line in the vertical direction.

Step 3: Measure the perpendicular distance of the lamp from the horizontal reference edge. Suppose this distance is $x$ cm.

Step 4: Measure the perpendicular distance of the lamp from the vertical reference edge. Suppose this distance is $y$ cm.

Step 5: The position of the table lamp can now be described to another person by saying: "The lamp is placed at a distance of $x$ cm from one edge (along the length) and $y$ cm from the adjacent edge (along the width) of the table."

Conclusion: Thus, using two perpendicular reference lines (the two adjacent edges of the table), the position of the lamp can be uniquely described by the ordered pair $(x, y)$.

Given:
- Two main roads cross at the centre of the city along North-South and East-West directions.
- All other streets are parallel to these roads and 200 m apart.
- There are 5 streets in each direction.
- Scale: 1 cm = 200 m.
- Convention: Cross-street $(m, n)$ means the crossing of the $m$-th North-South street and the $n$-th East-West street.

Model of the City:
Draw 5 vertical lines (representing North-South streets) and 5 horizontal lines (representing East-West streets), each 1 cm apart on the notebook. This gives a grid of $5 \times 5 = 25$ cross-streets.

Part (i): Cross-streets referred to as (4, 3)

The cross-street $(4, 3)$ is the point where the 4th North-South street meets the 3rd East-West street.

In the grid, there is exactly one 4th North-South street and exactly one 3rd East-West street. They can meet at only one point.

$$\boxed{\text{Only } 1 \text{ cross-street can be referred to as } (4, 3).}$$

Part (ii): Cross-streets referred to as (3, 4)

The cross-street $(3, 4)$ is the point where the 3rd North-South street meets the 4th East-West street.

Similarly, there is exactly one 3rd North-South street and exactly one 4th East-West street. They meet at only one point.

$$\boxed{\text{Only } 1 \text{ cross-street can be referred to as } (3, 4).}$$

Note: $(4, 3)$ and $(3, 4)$ refer to different cross-streets because the first number denotes the North-South street and the second denotes the East-West street. Interchanging the numbers gives a different location.