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.

(i) Names of the horizontal and vertical lines:

The horizontal line is called the $x$-axis.

The vertical line is called the $y$-axis.

Together, they are called the coordinate axes.

(ii) Name of each part of the plane formed by these two lines:

The two coordinate axes divide the Cartesian plane into four parts. Each part is called a Quadrant.
- The four quadrants are named: First Quadrant (I), Second Quadrant (II), Third Quadrant (III), and Fourth Quadrant (IV), numbered in the anti-clockwise direction starting from the upper right part.

(iii) Name of the point where the two lines intersect:

The point where the $x$-axis and the $y$-axis intersect is called the Origin, usually denoted by the letter O.

Its coordinates are $(0, 0)$.

Note: The answers are read directly from Fig. 3.14 of the NCERT textbook. The standard coordinates given in the NCERT figure are used below.

Concept:
- The abscissa (x-coordinate) of a point is its distance from the $y$-axis.
- The ordinate (y-coordinate) of a point is its distance from the $x$-axis.
- Coordinates of a point are written as (abscissa, ordinate).

(i) Coordinates of B:

From the figure, point B lies at $x = -5$ and $y = 2$.
$$\text{Coordinates of B} = (-5,\ 2)$$

(ii) Coordinates of C:

From the figure, point C lies at $x = 5$ and $y = -5$.
$$\text{Coordinates of C} = (5,\ -5)$$

(iii) Point identified by the coordinates $(-3, -5)$:

From the figure, the point located at $(-3, -5)$ is point E.
$$\text{Point} = E$$

(iv) Point identified by the coordinates $(2, -4)$:

From the figure, the point located at $(2, -4)$ is point G.
$$\text{Point} = G$$

(v) Abscissa of the point D:

From the figure, point D lies at $x = 6$.
$$\text{Abscissa of D} = 6$$

(vi) Ordinate of the point H:

From the figure, point H lies at $y = -3$.
$$\text{Ordinate of H} = -3$$

(vii) Coordinates of the point L:

From the figure, point L lies at $x = 0$ and $y = 5$.
$$\text{Coordinates of L} = (0,\ 5)$$

(viii) Coordinates of the point M:

From the figure, point M lies at $x = -3$ and $y = 0$.
$$\text{Coordinates of M} = (-3,\ 0)$$