Exercise 3.1: Coordinate Geometry — NCERT Solutions

(i) Origin: The point where the x-axis and the y-axis intersect. Its coordinates are $(0, 0)$.

(ii) Abscissa: The x-coordinate of a point; it is the perpendicular distance of the point from the y-axis, measured along the x-axis.

(iii) Ordinate: The y-coordinate of a point; it is the perpendicular distance of the point from the x-axis, measured along the y-axis.

(iv) Quadrant: The two axes divide the plane into four parts called quadrants — numbered I, II, III and IV in the anticlockwise direction starting from the upper-right region.

Use the sign rule: I $(+,+)$, II $(-,+)$, III $(-,-)$, IV $(+,-)$.

• $A(3,5)$: both positive → Quadrant I
• $B(-4,2)$: $(-,+)$ → Quadrant II
• $C(-6,-1)$: $(-,-)$ → Quadrant III
• $D(5,-3)$: $(+,-)$ → Quadrant IV
• $E(0,4)$: x-coordinate is $0$ → lies on the y-axis
• $F(-2,0)$: y-coordinate is $0$ → lies on the x-axis

To plot a point $(x, y)$, move $x$ units horizontally (right if positive, left if negative) from the origin, then $y$ units vertically (up if positive, down if negative).

• $P(2,3)$: 2 right, 3 up → Quadrant I
• $Q(-3,2)$: 3 left, 2 up → Quadrant II
• $R(-1,-4)$: 1 left, 4 down → Quadrant III
• $S(4,-2)$: 4 right, 2 down → Quadrant IV

Each point sits in a different quadrant, one in each region of the plane.

(i) On the x-axis, the y-coordinate is $0$. 'Left of origin' means negative x. So the point is $(-5, 0)$.

(ii) On the y-axis, the x-coordinate is $0$. 'Below the origin' means negative y. So the point is $(0, -3)$.

The four vertices are $O(0,0)$, $A(3,0)$, $B(3,4)$, $C(0,4)$.

Plotting them: $O$ and $A$ lie on the x-axis (3 units apart), $O$ and $C$ lie on the y-axis (4 units apart), and all angles are right angles. The figure is a rectangle.

• Length (along x-axis) $= 3$ units
• Breadth (along y-axis) $= 4$ units

$\text{Area} = \text{length} \times \text{breadth} = 3 \times 4 = 12 \text{ square units}$

For the point $(-7, 6)$:

• Abscissa (x-coordinate) $= -7$
• Ordinate (y-coordinate) $= 6$

Reflection in the y-axis: The y-axis reflection keeps the ordinate the same and changes the sign of the abscissa.

$(-7, 6) \;\longrightarrow\; (7, 6)$

So the mirror image is $(7, 6)$.

• $(-5, -8)$: abscissa $= -5$ → negative; ordinate $= -8$ → negative
• $(0, 7)$: abscissa $= 0$ → zero; ordinate $= 7$ → positive
• $(6, 0)$: abscissa $= 6$ → positive; ordinate $= 0$ → zero

Note that $(0,7)$ lies on the y-axis and $(6,0)$ lies on the x-axis.