What pattern do you observe when points share the same x-coordinate in Exercise 5.3?

When all points in a set share the same x-coordinate — for example (1,2), (1,3), (1,−4), (1,0), (1,8) — they all lie on a single vertical straight line that is parallel to the y-axis. The line passes through x = 1 at a horizontal distance of 1 unit from the y-axis. Similarly, points sharing the same y-coordinate all lie on a horizontal line parallel to the x-axis.

How do you calculate the area of a rectangle plotted on the Cartesian plane?

To find the area of a rectangle with vertices at (0,0), (0,3), (4,3), (4,0): first find the length by calculating the horizontal distance = 4 − 0 = 4 units, then find the width as the vertical distance = 3 − 0 = 3 units. Area = length × width = 4 × 3 = 12 square units. In general, you count grid units between the x-coordinates for length and between y-coordinates for width.

How is the area of a triangle calculated from its plotted vertices in Class 9 Exercise 5.3?

For the triangle with vertices (2,3), (6,3), and (4,7): the base is the horizontal distance between (2,3) and (6,3) = 6 − 2 = 4 units. The height is the perpendicular distance from the apex (4,7) to the base line y = 3, which equals 7 − 3 = 4 units. Area = ½ × base × height = ½ × 4 × 4 = 8 square units.

What does Question 7 of Exercise 5.3 reveal about points where x + y = 5?

Question 7 asks you to plot six points where the sum of coordinates equals 5. The points include (−2,7), (0,5), (1,4), (3,2), (5,0), and (6,−1). When plotted, all these points lie on a single straight line. This is the first introduction to the key concept of Class 9 Chapter 6: a linear equation in two variables — here x + y = 5 — produces a straight line graph in the Cartesian plane.

Exercise 5.3 — Plotting Points | Class 9 Maths | Co-ordinate Geometry

Plotting Points in the Cartesian Plane — Exercise 5.3

Exercise 5.3 is the practical application of everything learned in Exercises 5.1 and 5.2. Here you move from theory to action — actually plotting points, drawing shapes, identifying geometric patterns, and reading coordinates from a given graph. Every question in this exercise builds a different skill: reading a table of coordinates, recognising collinear points, computing area from plotted vertices, and inferring a figure from a set of line segments.

How to plot a point (x, y) step by step:
1. Start at the origin O.
2. Move |x| units to the right (if x > 0) or to the left (if x < 0).
3. From that position, move |y| units upward (if y > 0) or downward (if y < 0).
4. Mark the point and label it with its coordinates.

Question 1 — Plotting Six Points from a Table

The table below gives six pairs of x and y values. Form the ordered pairs and plot them on the Cartesian plane. The x-coordinate and y-coordinate together fix each point's position uniquely.

x	2	3	−1	0	−9	−4
y	−3	−3	4	11	0	−6
(x, y)	(2, −3)	(3, −3)	(−1, 4)	(0, 11)	(−9, 0)	(−4, −6)
Quadrant	Q₄	Q₄	Q₂	y-axis	x-axis	Q₃

Figure 1: Six points plotted from Question 1. Dashed lines show the reference projections to both axes.

Question 2 — Are (5, −8) and (−8, 5) the Same Point?

Question 2 — Analysis

Point A = (5, −8): x = 5 > 0, y = −8 < 0 → Fourth Quadrant (Q₄)

Point B = (−8, 5): x = −8 < 0, y = 5 > 0 → Second Quadrant (Q₂)

These are in different quadrants → they are different points

Ans: No. (5, −8) and (−8, 5) are different points in different quadrants. The order of coordinates matters — swapping them gives a completely different location.

Key Rule — Order of coordinates: In an ordered pair (x, y), the abscissa always comes first. Changing the order gives a different point. (5, −8) and (−8, 5) are mirror-like reflections of each other across the line y = x, but they occupy completely different positions in the plane.

Question 3 — Points Sharing the Same x-coordinate

Plot: (1,2), (1,3), (1,−4), (1,0), (1,8). Each point has x = 1.

Figure 2: All five points with x = 1 lie on the dashed vertical line — parallel to the y-axis at distance 1 unit.

Observation: All points (1,2), (1,3), (1,−4), (1,0), (1,8) lie on a single vertical line that is parallel to the y-axis at a distance of 1 unit from it. This confirms: points sharing the same x-coordinate form a vertical line.

Question 4 — Points Sharing the Same y-coordinate

Plot: (5,4), (8,4), (3,4), (0,4), (−4,4), (−2,4). Each point has y = 4.

Observation: All six points lie on a single horizontal line parallel to the x-axis at a distance of 4 units above it. Points sharing the same y-coordinate always form a horizontal line.

Question 5 — Rectangle from Four Vertices

Plot the points (0,0), (0,3), (4,3), (4,0) and join them to form a rectangle. Then calculate the area.

Figure 3: Rectangle with vertices (0,0), (0,3), (4,3), (4,0). Length = 4 units, Width = 3 units.

Calculation: Length of rectangle = distance from (0,0) to (4,0) = 4 units. Width = distance from (0,0) to (0,3) = 3 units. Area = length × width = 4 × 3 = 12 square units.

Question 6 — Triangle from Three Vertices

Plot (2,3), (6,3), (4,7) and find the area of the triangle formed.

Figure 4: Triangle with vertices (2,3), (6,3), (4,7). Base = 4 units, Height = 4 units.

Calculation: Base = distance between (2,3) and (6,3) = 6 − 2 = 4 units. Height = vertical distance from (4,7) to the base line y = 3 = 7 − 3 = 4 units. Area = ½ × base × height = ½ × 4 × 4 = 8 square units.

Question 7 — Six Points with Coordinate Sum = 5

Find and plot at least six points where x + y = 5. Some possibilities from the PDF:

Points where x + y = 5

(−2, 7): −2 + 7 = 5 ✓ → Q₂ (0, 5): 0 + 5 = 5 ✓ → y-axis (1, 4): 1 + 4 = 5 ✓ → Q₁ (3, 2): 3 + 2 = 5 ✓ → Q₁ (5, 0): 5 + 0 = 5 ✓ → x-axis (6, −1): 6 + (−1) = 5 ✓ → Q₄

All six points lie on a straight line: x + y = 5

Pattern: All points satisfying x + y = 5 lie on a single straight line. This is your first encounter with the concept that a linear equation in two variables produces a straight line when graphed on the Cartesian plane — a major theme in Chapter 6.

Question 8 — Reading All Coordinates from a Graph

Given a complex graph with 17 labelled points (A through Q), read and record each coordinate pair. The answers from the textbook are:

Point	Coordinates	Point	Coordinates
A	(−3, 4)	B	(0, 5)
C	(3, 4)	D	(2, 4)
E	(2, 0)	F	(3, 0)
G	(3, −1)	H	(0, −1)
I	(−3, −1)	J	(−3, 0)
K	(−2, 0)	L	(−2, 4)
M	(−1, 0)	N	(−1, 3)
O	(0, 0)	P	(1, 3)
Q	(1, 0)

Question 9 — Connecting Points to Form a Cupboard Shape

Fourteen pairs of points are given. Connect each pair with a straight line segment. When all 14 segments are drawn, the combined figure takes the shape of a cupboard. This question reinforces the practical use of plotting — geometric figures are simply collections of line segments defined by their endpoint coordinates.

Observation from Question 9: The shape of a cupboard is formed when all fourteen line segments are drawn in the correct sequence. This shows how complex real-world shapes can be precisely described and reproduced using nothing but a set of coordinate pairs.

Question 10 — Nine Intersecting Line Segments

Nine pairs of axis-intercept points are plotted and connected: (1,0)↔(0,9), (2,0)↔(0,8), ... , (9,0)↔(0,1). Each pair sums to 10 (the point on x-axis plus the y-intercept). The resulting pattern of nine lines creates a smooth curved appearance — a visual phenomenon called an envelope of a curve.

Each pair of points has coordinates summing to 10: e.g. (3,0) and (0,7) → x + y = 10 on that line.
The nine line segments, though straight, together approximate a curved shape (parabola-like arc).
This is a classic string-art or cardioid-envelope construction often used in art and mathematics demonstrations.