A complete visual guide — from the Cartesian plane and quadrants to plotting points, reading coordinates, and solving all exercises. Fully illustrated with interactive SVG diagrams.
Cartesian Plane
4 Quadrants
Abscissa & Ordinate
All Examples Solved
50 Practice Questions
Section 3.1 – 3.2
The Cartesian System
Invented by René Déscartes (1596–1650), this system locates any point in a plane using two perpendicular number lines.
Who Invented It?
René Déscartes, the great French mathematician and philosopher, invented Coordinate Geometry while lying in bed! He noticed a fly on the ceiling and thought about how to describe its exact position — he realised he needed its distance from two walls. This simple idea gave birth to Coordinate Geometry. The system is also called the Cartesian system in his honour.
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The word abscissa comes from Latin meaning "cut off" (x-distance). Ordinate comes from Latin meaning "ordered" (y-distance).
Why Do We Need Two Numbers?
One number is never enough to locate a point on a flat surface. Consider:
Situation
First Reference
Second Reference
Finding a house
Street number (East-West)
House number (North-South)
Dot on paper
Distance from left edge (5 cm)
Distance from bottom (9 cm)
Classroom seat
Column number (horizontal)
Row number (vertical)
Map location
Longitude (East-West)
Latitude (North-South)
Key insight: We always need two independent pieces of information to uniquely fix a position in a plane.
The Cartesian Plane — Fully Labelled
Key Definitions
x-axis (X'X): The horizontal number line. Positive direction → right (OX), negative direction → left (OX'). y-axis (Y'Y): The vertical number line. Positive direction → upward (OY), negative direction → downward (OY'). Origin (O): The point where the x-axis and y-axis intersect. Coordinates = (0, 0). Coordinate axes: The x-axis and y-axis together are called the coordinate axes. Cartesian plane / xy-plane: The plane formed by the two coordinate axes.
Section 3.2
The Four Quadrants
The two axes divide the entire plane into four equal parts called quadrants, numbered anticlockwise from OX.
Sign of Coordinates in Each Quadrant
Quadrant
x-coordinate (abscissa)
y-coordinate (ordinate)
Form
Example
I (First)
Positive (+)
Positive (+)
(+, +)
(3, 5)
II (Second)
Negative (−)
Positive (+)
(−, +)
(−2, 4)
III (Third)
Negative (−)
Negative (−)
(−, −)
(−5, −3)
IV (Fourth)
Positive (+)
Negative (−)
(+, −)
(6, −1)
On x-axis
Any value
Always 0
(x, 0)
(4, 0)
On y-axis
Always 0
Any value
(0, y)
(0, −3)
Origin
0
0
(0, 0)
O
🧠
Memory trick: Start in Quadrant I (both positive), go anticlockwise. Signs go: ++, −+, −−, +−. Remember: "All Students Take Calculus" — All (+,+), Sin (−,+) means y>0, Tan (−,−), Cos (+,−) means x>0. (This matches the ASTC rule for trigonometry too!)
Section 3.2
Understanding Coordinates
Every point in the plane has exactly one pair of coordinates. Learn the exact meaning of abscissa and ordinate.
Definition — Coordinates of a Point
x-coordinate (Abscissa): The perpendicular distance of a point from the y-axis, measured along the x-axis. Positive if to the right, negative if to the left.
y-coordinate (Ordinate): The perpendicular distance of a point from the x-axis, measured along the y-axis. Positive if upward, negative if downward.
Coordinates: Written as (x, y) — abscissa first, ordinate second. The pair (x, y) locates the point uniquely in the plane.
Method
How to Read Coordinates of a Point
1
Draw a perpendicular from the point to the x-axis. The number where it meets the x-axis is the x-coordinate (abscissa).
2
Draw a perpendicular from the point to the y-axis. The number where it meets the y-axis is the y-coordinate (ordinate).
3
Write the coordinates as (x-coordinate, y-coordinate) — x always comes first.
⚠️
Critical: (3, 4) and (4, 3) are different points! The order matters — x always comes before y. This is called an ordered pair.
Its x-coordinate = 0 (zero distance from y-axis). Its y-coordinate = 0 (zero distance from x-axis). Coordinates of O = (0, 0)
Key Rules to Remember
1. If x ≠ y, then (x, y) ≠ (y, x) — order matters in coordinates.
2. If x = y, then (x, y) = (y, x) — only when both values are equal.
3. A point on the x-axis: form (x, 0). Distance from x-axis = 0.
4. A point on the y-axis: form (0, y). Distance from y-axis = 0.
5. The origin is the only point that lies on both axes simultaneously.
Section 3.2
Plotting Points in the Cartesian Plane
Given (x, y), find its exact location on the grid — the fundamental skill of coordinate geometry.
Step-by-Step Method
How to Plot a Point P(4, 3)
1
Start at the origin O.
2
Move 4 units to the right along the x-axis (since x = +4). Mark this as M on the x-axis.
3
From M, move 3 units upward (since y = +3). Mark this point as P.
4
P is the point (4, 3). The perpendicular from P to x-axis = 3 (ordinate), perpendicular from P to y-axis = 4 (abscissa).
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Plotting directions summary: x positive → go RIGHT. x negative → go LEFT. y positive → go UP. y negative → go DOWN. Always start from the origin!
Textbook Examples
All Examples — Fully Solved
Every NCERT example from Chapter 3, with detailed step-by-step explanations.
Example 1
See Fig. 3.11 and find coordinates of points B, M, L and S.
B
Point B is 4 units right of y-axis → abscissa = 4. B is 3 units above x-axis → ordinate = 3. Coordinates of B = (4, 3)
M
Point M is 3 units left of y-axis → x-coordinate = –3. M is 4 units above x-axis → y-coordinate = 4. Coordinates of M = (–3, 4)
L
Point L is 5 units left of y-axis → x-coordinate = –5. L is 4 units below x-axis → y-coordinate = –4. Coordinates of L = (–5, –4)
S
Point S is 3 units right of y-axis → x-coordinate = 3. S is 4 units below x-axis → y-coordinate = –4. Coordinates of S = (3, –4)
B(4,3) · M(−3,4) · L(−5,−4) · S(3,−4)
Example 2
Write the coordinates of the points marked on the axes in Fig. 3.12. Points: A, B, C, D, E on axes.
A
A is on the x-axis, 4 units to the right of origin. Every point on x-axis has y = 0. Coordinates of A = (4, 0)
B
B is on the y-axis, 3 units above the origin. Every point on y-axis has x = 0. Coordinates of B = (0, 3)
C
C is on the x-axis, 5 units to the left of origin → x = −5, y = 0. Coordinates of C = (−5, 0)
D
D is on the y-axis, 4 units below the origin → x = 0, y = −4. Coordinates of D = (0, −4)
E
E is on the x-axis at 2/3 to the right of origin → x = 2/3, y = 0. Coordinates of E = (2/3, 0)
A(4,0) · B(0,3) · C(−5,0) · D(0,−4) · E(2/3, 0)
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Key rule: Every point on the x-axis has y-coordinate = 0. Every point on the y-axis has x-coordinate = 0. The origin is (0, 0) — it's on both axes.
NCERT Exercises
Exercise 3.1 & 3.2 — Fully Solved
Every question from both exercises, solved step by step.
Exercise 3.1
Q1
How will you describe the position of a table lamp on your study table to another person?
1
Choose two reference lines (edges of the table) perpendicular to each other — say the left edge and the bottom edge.
2
Measure the distance of the lamp from the left edge (say 25 cm) and from the bottom edge (say 30 cm).
3
Describe the position as: "The lamp is 25 cm from the left edge and 30 cm from the bottom edge of the table."
The position is described by two measurements from two perpendicular reference edges. Example: (25 cm from left, 30 cm from bottom).
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This is the real-life version of coordinates! The left edge acts as the y-axis, and the bottom edge acts as the x-axis.
Q2
(Street Plan) A city has two main roads along North-South and East-West. 5 streets in each direction, 200 m apart. Find: (i) How many cross-streets as (4,3)? (ii) How many as (3,4)?
1
The cross-street (x, y) = (street in N-S direction number x) meeting (street in E-W direction number y). Each pair of one N-S street and one E-W street meets at exactly ONE cross-street.
i
Cross-street (4, 3): The 4th N-S street meets the 3rd E-W street at exactly one point. So only 1 cross-street can be called (4, 3).
ii
Cross-street (3, 4): The 3rd N-S street meets the 4th E-W street at exactly one point. So only 1 cross-street can be called (3, 4). Note: (4, 3) and (3, 4) are different locations!
(i) Only 1 cross-street can be referred to as (4, 3). (ii) Only 1 cross-street can be referred to as (3, 4).
Exercise 3.2
Q1
Write the answer: (i) Name of horizontal and vertical lines? (ii) Name of each part of the plane? (iii) Name of the intersection point?
i
The horizontal line is called the x-axis. The vertical line is called the y-axis. Together they are called the coordinate axes.
ii
Each part of the plane formed by the two lines is called a quadrant. There are four quadrants, numbered I, II, III, IV anticlockwise from OX.
iii
The point where the two axes intersect is called the origin, denoted by O. Its coordinates are (0, 0).
(i) x-axis (horizontal), y-axis (vertical) (ii) Quadrants (iii) Origin O(0,0)
Q2
See Fig. 3.14 and write: (i) Coordinates of B (ii) Coordinates of C (iii) Point at (−3,−5) (iv) Point at (2,−4) (v) Abscissa of D (vi) Ordinate of H (vii) Coordinates of L (viii) Coordinates of M
i
B is at (−4, 2) — 4 units left of y-axis, 2 units above x-axis. Coordinates of B = (−4, 2)
ii
C is at (5, −5) — 5 units right, 5 units below. Coordinates of C = (5, −5)
iii
Point at (−3, −5): Look 3 left and 5 down → that's point E
iv
Point at (2, −4): 2 right and 4 down → that's point G
v
Abscissa (x-coordinate) of D: D is at (6, 2) → Abscissa of D = 6
vi
Ordinate (y-coordinate) of H: H is at (−5, −3) → Ordinate of H = −3
vii
L is 1 unit right of y-axis and 5 units above x-axis. Coordinates of L = (1, 5)
viii
M is on the x-axis, 3 units left → x = −3, y = 0. Coordinates of M = (−3, 0)
Master Coordinate Geometry smartly — avoid the most common mistakes students make in exams.
1
x ALWAYS Before y
Coordinates are written as (x, y) — never (y, x). The x-coordinate (abscissa) always comes first. (3, 4) ≠ (4, 3). This is the single most common mistake!
2
Learn the Sign Pattern
Q1:(+,+), Q2:(−,+), Q3:(−,−), Q4:(+,−). Anticlockwise. Memory aid: "ASTC" — All, Sin, Tan, Cos (same as trig quadrant signs).
3
Axes Are Special
Points on x-axis: y = 0. Points on y-axis: x = 0. Origin = (0,0). Don't put a point on the axis inside a quadrant — axes are boundaries, not quadrant points.
4
Abscissa = x, Ordinate = y
Abscissa = x-coordinate = distance from y-axis. Ordinate = y-coordinate = distance from x-axis. Many students mix these up in exams!
5
Always Use Graph Paper
When plotting points, always use proper graph paper or draw a neat grid. Mark the scale clearly. A rough sketch without proper scale loses marks.
6
Quadrant Number = Anticlockwise
Quadrants are numbered I, II, III, IV going ANTICLOCKWISE from the positive x-axis. Not clockwise! This is standard convention worldwide.
7
Origin Is On Both Axes
The origin (0, 0) lies on BOTH the x-axis and y-axis. It is NOT in any quadrant — it's at the junction of all four. Points ON the axes are also not in any quadrant.
8
Read x, then y, Plot!
To plot (a, b): Start at origin → move 'a' units along x-axis (right if +, left if −) → then move 'b' units vertically (up if +, down if −). Never move diagonally.
9
Negative Means Direction
A negative x means the point is to the LEFT of the y-axis. A negative y means BELOW the x-axis. The sign gives the DIRECTION, the number gives the DISTANCE.
10
Verify by Drawing Perpendiculars
After plotting a point, verify by drawing perpendiculars to both axes. The foot on x-axis should give x-coordinate; foot on y-axis should give y-coordinate.
Quick Reference
Chapter 3 — Formula & Fact Sheet
Concept
Rule / Formula
Example
Coordinates
(x, y) = (abscissa, ordinate)
(3, −2): x=3, y=−2
x-coordinate (abscissa)
Distance from y-axis, along x-axis
For (4,3): abscissa = 4
y-coordinate (ordinate)
Distance from x-axis, along y-axis
For (4,3): ordinate = 3
Origin
(0, 0)
O = (0,0)
Point on x-axis
(x, 0)
(5,0), (−3,0)
Point on y-axis
(0, y)
(0,4), (0,−2)
Quadrant I
(+, +): x>0, y>0
(2,5), (3,1)
Quadrant II
(−, +): x<0, y>0
(−3,4), (−1,2)
Quadrant III
(−, −): x<0, y<0
(−5,−2), (−1,−1)
Quadrant IV
(+, −): x>0, y<0
(4,−3), (6,−5)
Order matters
If x≠y, then (x,y) ≠ (y,x)
(3,4) ≠ (4,3)
Numbering
Quadrants I→IV: Anticlockwise from OX
Standard convention
CBSE Pattern Practice
50 Practice Questions
All CBSE question types — MCQ, 1-mark, 2-mark, 3-mark, and 5-mark questions with complete answers.
Section A — MCQ (1 mark each)
1
The point (−3, 4) lies in which quadrant?
MCQ
(a) Quadrant I
(b) Quadrant II
(c) Quadrant III
(d) Quadrant IV
Answer: (b) Quadrant II x = −3 (negative), y = 4 (positive) → (−, +) → Quadrant II
2
The coordinates of the origin are:
MCQ
(a) (1, 0)
(b) (0, 1)
(c) (0, 0)
(d) (1, 1)
Answer: (c) (0, 0) The origin O has zero distance from both axes. So both coordinates are 0.
3
A point on the y-axis has:
MCQ
(a) x-coordinate = 0
(b) y-coordinate = 0
(c) both = 0
(d) neither = 0
Answer: (a) x-coordinate = 0 Every point on the y-axis is on the y-axis itself, so its distance from the y-axis (= x-coordinate) is 0. The form is (0, y).
4
The abscissa of the point (−5, 7) is:
MCQ
(a) 7
(b) −7
(c) 5
(d) −5
Answer: (d) −5 Abscissa = x-coordinate. For point (−5, 7), x-coordinate = −5.
5
The point (4, −3) lies in:
MCQ
(a) Q I
(b) Q II
(c) Q III
(d) Q IV
Answer: (d) Q IV x = +4, y = −3 → (+, −) → Quadrant IV
6
Which point lies on the x-axis?
MCQ
(a) (0, 3)
(b) (5, 0)
(c) (2, 2)
(d) (−3, −3)
Answer: (b) (5, 0) Points on the x-axis have y = 0. Only (5, 0) satisfies this.
7
The quadrants are numbered:
MCQ
(a) Clockwise from OX
(b) Anticlockwise from OX
(c) Clockwise from OY
(d) Randomly
Answer: (b) Anticlockwise from OX Quadrants are numbered I, II, III, IV in the anticlockwise direction starting from the positive x-axis (OX).
8
If coordinates of a point are (−2, −5), it lies in:
MCQ
(a) Q I
(b) Q II
(c) Q III
(d) Q IV
Answer: (c) Q III (−, −) → Quadrant III. Both coordinates negative → Third Quadrant.
9
The ordinate of the point (8, −11) is:
MCQ
(a) 8
(b) −8
(c) 11
(d) −11
Answer: (d) −11 Ordinate = y-coordinate. For (8, −11), y = −11.
10
Is (3, 4) the same as (4, 3) in the coordinate plane?
MCQ
(a) Yes, always
(b) No, never
(c) Sometimes
(d) Only if on axes
Answer: (b) No, never Since 3 ≠ 4, (3,4) ≠ (4,3). They represent completely different points in the plane. Order always matters in coordinates.
Section B — Very Short Answer (1 mark each)
11
Write the coordinates of a point that lies in Quadrant III and is 4 units from the y-axis and 3 units from the x-axis.
1 Mark
Answer: (−4, −3) Q III means x negative, y negative. 4 units from y-axis → x = −4. 3 units from x-axis → y = −3.
12
In which quadrant does the point (−1, 1) lie?
1 Mark
Answer: Quadrant II x = −1 (negative), y = 1 (positive) → (−, +) → Quadrant II
13
What is the name of the point where the x-axis and y-axis meet?
1 Mark
Answer: Origin (O) The point of intersection of the coordinate axes is called the origin, denoted O, with coordinates (0, 0).
14
Write the coordinates of a point on the y-axis at a distance of 5 units from the origin (below the x-axis).
1 Mark
Answer: (0, −5) On y-axis → x = 0. Below x-axis → y is negative. 5 units from origin below → y = −5. Point: (0, −5).
15
The point (−5, 0) lies on which axis?
1 Mark
Answer: x-axis (negative direction) y = 0 means the point is on the x-axis. x = −5 means it's to the left of the origin, on the negative x-axis.
16
What is the abscissa of all points on the y-axis?
1 Mark
Answer: 0 (zero) Every point on the y-axis has zero perpendicular distance from the y-axis. So abscissa (x-coordinate) = 0 for all points on y-axis.
17
The coordinate plane is divided into how many quadrants? Name them.
1 Mark
Answer: 4 quadrants Quadrant I (top-right), Quadrant II (top-left), Quadrant III (bottom-left), Quadrant IV (bottom-right). Numbered anticlockwise from OX.
18
Write the sign of x-coordinate and y-coordinate in Quadrant IV.
1 Mark
Answer: x positive (+), y negative (−) Quadrant IV is enclosed by positive x-axis and negative y-axis. So x > 0 and y < 0. Form: (+, −).
19
Give the full name of the system used to locate a point in a plane using two perpendicular axes.
1 Mark
Answer: Cartesian System (also called Rectangular Coordinate System or Coordinate Geometry system). Named after René Déscartes (1596–1650).
20
What are the coordinates of the point that is 3 units to the right of the origin on the x-axis?
1 Mark
Answer: (3, 0) On x-axis → y = 0. 3 units to the right → x = +3. Coordinates: (3, 0).
Section C — Short Answer (2 marks each)
21
In which quadrant or on which axis do the following points lie? A(2,3), B(−3,0), C(−4,−5), D(1,−2), E(0,−6), F(5,7).
2 Mark
A(2,3): (+,+) → Q I B(−3,0): y=0 → negative x-axis C(−4,−5): (−,−) → Q III D(1,−2): (+,−) → Q IV E(0,−6): x=0 → negative y-axis F(5,7): (+,+) → Q I
22
Write the coordinates of the points: P lies 4 units below the x-axis on the y-axis. Q lies 2 units to the left of origin on the x-axis. R lies in Q II at 5 units from y-axis and 3 units from x-axis.
2 Mark
P: On y-axis (x=0), 4 below (y=−4) → P = (0, −4) Q: On x-axis (y=0), 2 left (x=−2) → Q = (−2, 0) R: Q II means x<0, y>0. 5 from y-axis → x=−5. 3 from x-axis → y=3. R = (−5, 3)
23
Plot the points A(3,4), B(−3,4), C(−3,−4), D(3,−4) on the Cartesian plane. What shape do they form?
2 Mark
Plotting: A(3,4) in Q I, B(−3,4) in Q II, C(−3,−4) in Q III, D(3,−4) in Q IV. All four points have |x|=3 and |y|=4. Width = 6 units, height = 8 units. Shape: Rectangle (sides parallel to axes, opposite sides equal: AB=CD=6 units, BC=AD=8 units).
24
The x-coordinate of a point is −6 and its y-coordinate is the same as its x-coordinate. Write the point and state which quadrant it lies in.
2 Mark
Answer: x = −6. y = x = −6. So the point is (−6, −6). Both coordinates negative → (−, −) → Quadrant III
25
A point P is at a perpendicular distance of 7 units from the y-axis to its right, and 2 units above the x-axis. Write the coordinates of P and state its quadrant.
2 Mark
Answer: 7 units right of y-axis → x = +7. 2 units above x-axis → y = +2. Point P = (7, 2). Since x>0 and y>0 → Quadrant I.
26
Plot the points: O(0,0), A(4,0), B(4,3), C(0,3). What figure is formed by joining them in order?
2 Mark
O(0,0): Origin. A(4,0): on x-axis. B(4,3): in Q I. C(0,3): on y-axis. OA = 4 units (horizontal), AB = 3 units (vertical), BC = 4 units (horizontal), CO = 3 units (vertical). Figure: Rectangle with length 4 and width 3. Area = 12 sq units.
27
If a point lies on the negative direction of both axes, which quadrant does it lie in? Give an example.
2 Mark
Answer: Quadrant III Negative x (OX' direction) and negative y (OY' direction) → both coordinates negative → (−, −) → Quadrant III. Example: (−3, −5), (−1, −7), (−10, −2) — all in Q III.
28
Find the coordinates of the mirror image of the point (3, −5) in the x-axis.
2 Mark
Answer: (3, 5) Reflection in x-axis: keep the x-coordinate the same, change the sign of the y-coordinate. (3, −5) → mirror in x-axis → (3, +5) = (3, 5). The point moves from Q IV to Q I.
29
How many points on the number line can be represented as (x, 0)? Explain what the x-axis represents.
2 Mark
Answer: Infinitely many points. For every real number x, the point (x, 0) exists on the x-axis. Since there are infinitely many real numbers, there are infinitely many such points. The x-axis is itself a complete number line where every real number has a corresponding point of the form (x, 0).
30
Write the mirror image of (−4, 6) in the y-axis and state which quadrant the image lies in.
2 Mark
Answer: (4, 6) — Quadrant I Reflection in y-axis: change the sign of x-coordinate, keep y-coordinate. (−4, 6) → mirror in y-axis → (+4, 6) = (4, 6). Since x>0 and y>0 → Quadrant I.
Section D — Short Answer II (3 marks each)
31
Plot the points A(1,1), B(4,1), C(4,4), D(1,4) on a graph. Join them in order. What figure is formed? Find its perimeter and area.
3 Mark
All four points are in Q I. AB: from (1,1) to (4,1) = 3 units (horizontal) BC: from (4,1) to (4,4) = 3 units (vertical) CD: from (4,4) to (1,4) = 3 units (horizontal) DA: from (1,4) to (1,1) = 3 units (vertical) Figure: Square with side 3 units. Perimeter = 4 × 3 = 12 units. Area = 3² = 9 sq units.
32
In Fig. 3.14 (refer to Exercise 3.2), write all the points visible and state the quadrant or axis for each: B, C, D, E, G, H, L, M.
3 Mark
B(−4,2): Q II | C(5,−5): Q IV | D(6,2): Q I | E(−3,−5): Q III | G(4,−4): Q IV | H(−5,−3): Q III | L(1,5): Q I | M(−3,0): negative x-axis (on axis, not in quadrant)
33
A point P(a, b) satisfies a = −b. In which quadrant(s) can P lie? Give one example for each possible quadrant.
3 Mark
Condition: a = −b → b = −a. If a > 0: b = −a < 0 → (+, −) → Q IV. Example: (3, −3). If a < 0: b = −a > 0 → (−, +) → Q II. Example: (−2, 2). If a = 0: b = 0 → point is origin (0, 0) — on axes, no quadrant. P can lie in Q II or Q IV (or at origin).
34
Without plotting, determine the quadrant of: A(15,−3), B(−√2, −7), C(0.5, 0.5), D(−100, 200), E(1/3, −2/5).
3 Mark
A(15,−3): x>0, y<0 → Q IV B(−√2,−7): x<0 (−√2≈−1.41), y<0 → Q III C(0.5,0.5): x>0, y>0 → Q I D(−100,200): x<0, y>0 → Q II E(1/3,−2/5): x>0 (1/3>0), y<0 (−2/5<0) → Q IV
35
The vertices of a triangle are A(2,4), B(6,4), C(4,7). Plot these points and describe the type of triangle formed.
3 Mark
All points in Q I. AB: both have y=4, so AB is horizontal. AB = |6−2| = 4 units. M (midpoint of AB) = (4, 4). C = (4,7). CM is vertical, length = 3 units. AC = √((4−2)²+(7−4)²) = √(4+9) = √13. BC = √((4−6)²+(7−4)²) = √13. AC = BC → Isosceles triangle. Also AB is base, CM is perpendicular bisector → axis of symmetry.
36
Point A is in Q II with abscissa −5. Point B is the reflection of A in the x-axis and lies in Q III. Is this statement correct? If abscissa of A is −5 and ordinate is 3, find coordinates of A, B, and the mirror image of A in the y-axis.
3 Mark
A(−5, 3): x<0, y>0 → Q II ✓ Reflection of A in x-axis: Change sign of y → (−5, −3). x<0, y<0 → Q III ✓ So B = (−5, −3). The statement "B lies in Q III" is correct! ✓ Mirror image of A in y-axis: Change sign of x → (5, 3). This is in Q I.
37
A ship is at point S(−4, 3) on a map. A lighthouse is at L(4, 3). Find the horizontal distance between them (both points are at same height/y-coordinate). What type of line segment SL is?
3 Mark
S is at (−4, 3) and L is at (4, 3). Both have y = 3 → they are at the same height (same y-coordinate). Horizontal distance = |4 − (−4)| = |4 + 4| = 8 units. Since y is same for both points, SL is a horizontal line segment, parallel to the x-axis.
38
Plot 4 points, one in each quadrant, such that each point is equidistant (distance 5 units) from the origin. State their coordinates.
3 Mark
We need points (x,y) where |x|² + |y|² = 25. Using x=3, y=4: Q I: (3, 4) — distance from O = √(9+16) = √25 = 5 ✓ Q II: (−3, 4) — distance = 5 ✓ Q III: (−3, −4) — distance = 5 ✓ Q IV: (3, −4) — distance = 5 ✓ All four points are 5 units from the origin (they lie on a circle of radius 5).
39
A point P(x, y) is such that xy > 0. In which quadrant(s) can it lie? What if xy < 0?
3 Mark
xy > 0 means x and y have the SAME sign: Both positive → Q I (+,+) where xy>0 ✓ Both negative → Q III (−,−) where xy = (neg)(neg) = positive ✓ → P lies in Q I or Q III when xy > 0
xy < 0 means x and y have OPPOSITE signs: x>0, y<0 → Q IV ✓ x<0, y>0 → Q II ✓ → P lies in Q II or Q IV when xy < 0
40
In the Seating Plan activity, a student sits in the 4th column and 3rd row. Another sits in the 3rd column and 4th row. Are these the same position? Explain using coordinate geometry concepts.
3 Mark
Student 1: Column 4, Row 3 → Position (4, 3). Student 2: Column 3, Row 4 → Position (3, 4). These are NOT the same position. Just as (4, 3) ≠ (3, 4) in the coordinate plane (since 4 ≠ 3), the two seating positions are different. The column number is the x-coordinate and the row number is the y-coordinate. Order matters — always state column (x) first, then row (y).
Section E — Long Answer (5 marks each)
41
Draw a Cartesian plane with all labels. Plot the following points: A(4,3), B(−3,4), C(−5,−2), D(3,−4), E(0,5), F(−4,0), G(0,0). State the quadrant or axis for each point.
5 Mark
Drawing: Draw x-axis (horizontal), y-axis (vertical), label X, X', Y, Y', mark units, label O. A(4,3): 4 right, 3 up → Q I B(−3,4): 3 left, 4 up → Q II C(−5,−2): 5 left, 2 down → Q III D(3,−4): 3 right, 4 down → Q IV E(0,5): on positive y-axis F(−4,0): on negative x-axis G(0,0): origin O
42
Describe with examples: (a) What are coordinates? (b) Why is order important? (c) What are the coordinates on each axis and at origin? (d) What do the signs of coordinates tell us?
5 Mark
(a) Coordinates: An ordered pair (x, y) that gives the exact position of a point. x = perpendicular distance from y-axis, y = perpendicular distance from x-axis. Example: (3, −2) is 3 right and 2 below origin.
(b) Order matters: (3,4) ≠ (4,3). Example: A(3,4) is in Q I but B(4,3) is also in Q I — they are different points! A is 3 right, 4 up; B is 4 right, 3 up. Coordinates are ordered pairs.
Plot the points A(2,3), B(−2,3), C(−2,−3), D(2,−3). Join ABCD. What figure is formed? State all its properties in terms of coordinates.
5 Mark
Points: A(2,3) in QI, B(−2,3) in QII, C(−2,−3) in QIII, D(2,−3) in QIV. Figure formed: Rectangle ABCD AB: y=3 for both → horizontal. Length = |2−(−2)| = 4 units. BC: x=−2 for both → vertical. Length = |3−(−3)| = 6 units. CD: y=−3 for both → horizontal. Length = 4 units. DA: x=2 for both → vertical. Length = 6 units. Properties: All angles 90°. Opposite sides equal (4 and 6). Sides parallel to axes. Symmetric about both axes. Perimeter = 20. Area = 24 sq units. Diagonals equal and bisect each other at origin (midpoint check: midpoint AC = (0,0), midpoint BD = (0,0) ✓).
44
The Cartesian plane is like a city map. Explain with a real-life analogy: (a) what the axes represent (b) what the quadrants represent (c) what coordinates represent (d) why order of coordinates matters (e) what the origin represents.
5 Mark
(a) Axes = Main Roads. The two main perpendicular roads (E-W and N-S) of a city are like the x-axis and y-axis. All other streets run parallel to these.
(b) Quadrants = City Sectors. The two main roads divide the city into 4 sectors (NE, NW, SW, SE), just like the axes divide the plane into 4 quadrants.
(c) Coordinates = Address. "Street 3 East, 5 blocks North" = (3, 5). Two numbers uniquely identify any location in the city (and any point in the plane).
(d) Order matters = Street, then block number. "Street 3, House 5" is different from "Street 5, House 3" — they're at different locations. Similarly (3,5) ≠ (5,3).
(e) Origin = Town Square / City Centre. The central meeting point from which all distances are measured. Every distance in the city is measured from this point.
45
From the graph in Exercise 3.2 (Fig 3.14), write all given information: coordinates of all visible points, which quadrant/axis each lies in, and identify any points on the axes.
5 Mark
From Fig 3.14: L(1,5) — Q I | B(−4,2) — Q II | D(6,2) — Q I | M(−3,0) — negative x-axis O(0,0) — Origin | H(−5,−3) — Q III | G(4,−4) — Q IV | E(−3,−5) — Q III | C(5,−5) — Q IV Points on axes: M(−3,0) on negative x-axis, O(0,0) at origin. Points in each quadrant: QI: L, D. QII: B. QIII: H, E. QIV: G, C. Total: 2 in QI, 1 in QII, 2 in QIII, 2 in QIV, 1 on axis, 1 at origin.
46
Plot the following points and connect them in order. Name the figure: P(0,4), Q(3,0), R(0,−4), S(−3,0). Find the perimeter of the figure.
5 Mark
Points: P(0,4) on +y-axis, Q(3,0) on +x-axis, R(0,−4) on −y-axis, S(−3,0) on −x-axis. Figure: Rhombus (all sides equal, diagonals along the axes). PQ = √((3−0)²+(0−4)²) = √(9+16) = √25 = 5 units. QR = √((0−3)²+(−4−0)²) = √(9+16) = 5 units. RS = √((−3−0)²+(0+4)²) = 5 units. SP = √((0+3)²+(4−0)²) = 5 units. Perimeter = 4 × 5 = 20 units. Diagonals: PR = 8 units (vertical), QS = 6 units (horizontal), perpendicular to each other → confirms rhombus.
47
Write a comprehensive note on René Déscartes and his contribution to mathematics. How does the Cartesian system help in real life? Give 5 real-world applications.
5 Mark
René Déscartes (1596–1650): French philosopher and mathematician. His famous saying "I think, therefore I am" made him one of history's greatest thinkers. He invented the Cartesian coordinate system while observing a fly on a ceiling — wondering how to describe its exact position led him to the two-axis system.
Contribution: He unified algebra and geometry by showing that any geometric shape can be described by algebraic equations. This gave rise to Coordinate Geometry (Analytic Geometry).
Real-life applications: 1. GPS Navigation: Your phone uses coordinates (latitude, longitude) to locate you — exactly like (x, y). 2. Computer graphics: Every pixel on a screen has coordinates (x, y). 3. Engineering & Architecture: Blueprints use coordinates to place every structural element precisely. 4. Maps: Grid references on maps use the same principle. 5. Video games: Character positions, movements, and collisions all use coordinate systems.
48
A point P(a, b) lies in Q II. State the signs of a and b. Now: (i) P moves 3 units right and 2 units down to reach Q. Write Q in terms of a and b. (ii) If Q lies on x-axis, find b. (iii) If additionally a = −5, find all coordinates.
5 Mark
P(a,b) in Q II → a < 0, b > 0
(i) Moving 3 right: new x = a+3. Moving 2 down: new y = b−2. So Q = (a+3, b−2).
(ii) Q on x-axis → y-coordinate = 0. b−2 = 0 → b = 2.
(iii) a = −5, b = 2. So P = (−5, 2). Q = (−5+3, 2−2) = (−2, 0). P(−5, 2) is in Q II ✓ (a=−5<0, b=2>0). Q(−2, 0) is on the negative x-axis ✓.
49
Explain all 11 summary points of Chapter 3 Coordinate Geometry in your own words, with one example for each point.
5 Mark
1. Need 2 ⊥ lines to locate a point in a plane. Ex: left edge + bottom edge of paper. 2. Plane = Cartesian / coordinate plane; lines = coordinate axes. 3. Horizontal = x-axis; Vertical = y-axis. Ex: x-axis is like East-West road. 4. Axes divide plane into 4 quadrants. Ex: Q I is top-right. 5. Intersection = origin O. Ex: O = (0,0). 6. x-coord = distance from y-axis (abscissa); y-coord = distance from x-axis (ordinate). Ex: For (3,4): abscissa=3, ordinate=4. 7. (x,y) = coordinates of the point. Ex: A(2,−3). 8. x-axis points: (x,0). y-axis points: (0,y). Ex: (5,0), (0,−2). 9. Origin = (0,0). Ex: O(0,0). 10. Q I:(+,+), Q II:(−,+), Q III:(−,−), Q IV:(+,−). Ex: (−2,3) in Q II. 11. If x≠y, (x,y)≠(y,x). If x=y, (x,y)=(y,x). Ex: (3,4)≠(4,3); (5,5)=(5,5).
50
Design a coordinate geometry puzzle: place 5 points A, B, C, D, E such that A is in Q I, B is in Q II, C is on the y-axis (positive), D is in Q IV, and E is at the origin. Then answer: (i) Which points have positive x? (ii) Which have negative y? (iii) Which are not in any quadrant? (iv) Sum of all x-coordinates if A=(3,2), B=(−4,1), C=(0,5), D=(2,−3), E=(0,0).