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1. Introduction

This chapter introduces coordinate geometry and the idea of representing points in a plane using coordinates.

• Coordinates help us locate points exactly in a plane.
• The system used is called the Cartesian plane, named after René Descartes.

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2. Cartesian Plane

• A Cartesian plane has:
  1. X-axis: horizontal line
  2. Y-axis: vertical line
  3. Origin (O): point where X-axis and Y-axis intersect (0, 0)
• The plane is divided into 4 quadrants:

Quadrant	X-coordinate	Y-coordinate
I	+	+
II	–	+
III	–	–
IV	+	–

Note: Coordinates are written as (x, y).

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3. Points and Coordinates

• A point on the plane is represented by a pair of numbers: $(x, y)$(x,y)
  • $x$x = distance from Y-axis (horizontal movement)
  • $y$y = distance from X-axis (vertical movement)

Examples:

1. $P(3, 4)$P(3,4): 3 units along X-axis, 4 units along Y-axis
2. $Q(-2, 5)$Q(−2,5): 2 units left of Y-axis, 5 units above X-axis

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4. Plotting Points

Steps to plot a point (x, y):

1. Start at the origin (0,0)
2. Move x units along X-axis: right if positive, left if negative
3. Move y units along Y-axis: up if positive, down if negative
4. Mark the point

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5. Quadrants

• Quadrant I: (+, +) → Right and Up
• Quadrant II: (-, +) → Left and Up
• Quadrant III: (-, -) → Left and Down
• Quadrant IV: (+, -) → Right and Down

Example: $(-3, -4)$(−3,−4) is in Quadrant III.

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6. Distance Between Points

• Horizontal distance between $(x_1, y_1)$(x1​,y1​) and $(x_2, y_1)$(x2​,y1​) = $|x_2 – x_1|$∣x2​−x1​∣
• Vertical distance between $(x_1, y_1)$(x1​,y1​) and $(x_1, y_2)$(x1​,y2​) = $|y_2 – y_1|$∣y2​−y1​∣
• Distance formula (optional for reference):

$$\text{Distance} = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$Distance=(x2​−x1​)2+(y2​−y1​)2​

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7. Types of Points

1. Origin: (0, 0)
2. Points on X-axis: y = 0 (example: (5,0))
3. Points on Y-axis: x = 0 (example: (0, -3))
4. Points in Quadrants: coordinates according to the quadrant rules

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8. Simple Geometrical Figures on the Plane

• Shapes can be plotted using points:
  • Triangle: 3 points
  • Rectangle / Square: 4 points
  • Example: Rectangle with vertices $(1,1), (1,4), (5,4), (5,1)$(1,1),(1,4),(5,4),(5,1)
• Plot points and join them to visualize the shape.

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9. Reflection and Symmetry

• Reflection about X-axis: Change sign of y → $(x, y) → (x, -y)$(x,y)→(x,−y)
• Reflection about Y-axis: Change sign of x → $(x, y) → (-x, y)$(x,y)→(−x,y)

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10. Summary Table

Concept	Key Idea
Cartesian Plane	X-axis + Y-axis, origin O(0,0)
Coordinates	(x, y) – x: horizontal, y: vertical
Quadrants	I(+,+), II(-,+), III(-,-), IV(+,-)
Plotting a Point	Start at origin, move x then y, mark
Distance (horizontal)
Distance (vertical)
Reflection X-axis	(x, y) → (x, -y)
Reflection Y-axis	(x, y) → (-x, y)

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11. Key Tips

1. Always label axes clearly.
2. Check the sign of coordinates before plotting.
3. Use grid lines for accuracy.
4. Remember origin is (0,0).

Class 7 Maths – Chapter 3: A Peek Beyond the Point

50 Mixed-Type Questions

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Section A: Multiple Choice Questions (MCQs) – 10 Questions

1. The point where X-axis and Y-axis intersect is called:
   a) Quadrant I
   b) Origin
   c) Coordinate
   d) Axis
2. Which of the following points lies on the Y-axis?
   a) (0, 5)
   b) (3, 0)
   c) (-2, -3)
   d) (4, 2)
3. Point (–3, 4) lies in:
   a) Quadrant I
   b) Quadrant II
   c) Quadrant III
   d) Quadrant IV
4. If a point has coordinates (x, 0), it lies on:
   a) X-axis
   b) Y-axis
   c) Origin
   d) Quadrant I
5. The coordinates of the origin are:
   a) (1, 0)
   b) (0, 1)
   c) (0, 0)
   d) (1, 1)
6. Which quadrant has coordinates (+, –)?
   a) I
   b) II
   c) III
   d) IV
7. The reflection of (3, –4) about X-axis is:
   a) (3, 4)
   b) (–3, 4)
   c) (–3, –4)
   d) (3, –4)
8. The distance between points (0, 0) and (0, 5) is:
   a) 0
   b) 5
   c) √5
   d) 25
9. Which of the following points lies in Quadrant III?
   a) (–2, –3)
   b) (3, 2)
   c) (–1, 4)
   d) (5, –3)
10. If a rectangle has vertices at (1,1), (1,4), (5,4), and (5,1), the length along X-axis is:
    a) 3
    b) 4
    c) 5
    d) 1

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Section B: Fill in the Blanks – 10 Questions

11. The plane formed by X-axis and Y-axis is called the _______.
12. The first coordinate in (x, y) is called the _______.
13. The second coordinate in (x, y) is called the _______.
14. Point (0, 0) is called the _______.
15. A point (–4, 3) is in Quadrant _______.
16. Coordinates of a point on X-axis have _______ as their second number.
17. Coordinates of a point on Y-axis have _______ as their first number.
18. The reflection of (–2, 5) about Y-axis is _______.
19. The reflection of (3, –4) about X-axis is _______.
20. The horizontal distance between (2, 5) and (7, 5) is _______.

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Section C: Short Answer Questions – 10 Questions

21. Plot the points (2, 3), (–2, 3), (–2, –3), and (2, –3) and identify the quadrant of each.
22. Find the coordinates of a point 4 units to the right of the origin and 3 units above the X-axis.
23. Find the reflection of the point (5, –2) about X-axis.
24. Find the reflection of the point (–3, 6) about Y-axis.
25. A point lies 6 units left of Y-axis and 2 units above X-axis. Write its coordinates.
26. Name the quadrants of points (–4, 2), (3, –5), and (–1, –3).
27. Determine the vertical distance between points (5, 2) and (5, 8).
28. Determine the horizontal distance between points (–3, 7) and (2, 7).
29. What are the coordinates of the midpoint between points (2, 3) and (6, 7)?
30. If a point lies on the X-axis, its y-coordinate is _______.

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Section D: Long Answer / Problem-Solving – 10 Questions

31. Plot a triangle with vertices (1,1), (4,1), and (1,5). Find the lengths of its sides.
32. Plot a rectangle with vertices (–1,1), (3,1), (3,4), and (–1,4). Find its area.
33. A point P is at (3, 4). A point Q is at (3, –2). Find the vertical distance between P and Q.
34. The distance between points A(–2, –3) and B(–2, 2) is _______.
35. A square has vertices at (0,0), (0,3), (3,3), (3,0). Find its perimeter.
36. Find the coordinates of a point which is 5 units right and 4 units above the origin.
37. Plot points (1,2), (2,4), (3,6), (4,8) and check whether they lie on a straight line.
38. Find the midpoint of line segment joining (–3, –4) and (5, 2).
39. If a point (x, y) is reflected about X-axis, how do the coordinates change? Give one example.
40. A triangle has vertices (0,0), (0,3), and (4,0). Find its area.

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Section E: Application / Higher-Order Thinking – 10 Questions

41. A rectangle has vertices (–2,1), (2,1), (2,4), and (–2,4). Find its area and perimeter.
42. A point lies 3 units left of Y-axis and 7 units below X-axis. Write its coordinates and quadrant.
43. Reflect point (–4, –3) about Y-axis and X-axis, and write new coordinates.
44. Plot points (2,3), (–2,3), (–2,–3), (2,–3) and name the quadrants of each point.
45. A line segment has endpoints (–2, 5) and (4, 5). Find its length.
46. Find the distance between points (–3, 0) and (3, 0).
47. Midpoint of (–4, 6) and (2, –2) is _______.
48. Write coordinates of a point on the X-axis 7 units to the right of origin.
49. A rectangle’s vertices are (1,1), (1,5), (6,5), (6,1). Find its area.
50. Determine the quadrant for points (5,–3), (–7,4), (–2,–6), (8,9).

Answers – Chapter 3: A Peek Beyond the Point

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Section A: MCQs – Answers

1. b) Origin – Intersection of X-axis and Y-axis.
2. a) (0, 5) – On Y-axis, x = 0.
3. b) Quadrant II – x negative, y positive.
4. a) X-axis – y = 0 for all points on X-axis.
5. c) (0, 0) – Origin coordinates.
6. d) IV – x positive, y negative.
7. a) (3, 4) – Reflection about X-axis: y → -y.
8. b) 5 – Vertical distance = |5 – 0| = 5 units.
9. a) (–2, –3) – Quadrant III.
10. b) 4 – Length along X-axis = 5 – 1 = 4.

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Section B: Fill in the Blanks – Answers

11. Cartesian plane
12. x-coordinate / abscissa
13. y-coordinate / ordinate
14. Origin
15. II – x negative, y positive
16. 0 – y = 0 for points on X-axis
17. 0 – x = 0 for points on Y-axis
18. (2, 5) – Change sign of x (reflection about Y-axis)
19. (3, 4) – Change sign of y (reflection about X-axis)
20. 5 – Horizontal distance = |7 – 2| = 5

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Section C: Short Answer – Answers

21. (2,3) → Quadrant I; (–2,3) → Quadrant II; (–2,–3) → Quadrant III; (2,–3) → Quadrant IV
22. Coordinates = (4,3)
23. Reflection about X-axis: (5, 2) → (5, –2)
24. Reflection about Y-axis: (–3, 6) → (3, 6)
25. Coordinates = (–6, 2)
26. Quadrants: (–4,2) → II; (3,–5) → IV; (–1,–3) → III
27. Vertical distance = |8 – 2| = 6
28. Horizontal distance = |2 – (–3)| = 5
29. Midpoint = ((2+6)/2, (3+7)/2) = (4, 5)
30. y = 0

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Section D: Long Answer / Problem-Solving – Answers

31. Triangle sides:

• Base: 4 – 1 = 3 units
• Height: 5 – 1 = 4 units
• Hypotenuse: √[(4–1)² + (5–1)²] = √[3² + 4²] = 5 units

32. Rectangle:

• Length = 3 – (–1) = 4
• Width = 4 – 1 = 3
• Area = 4 × 3 = 12 units²

33. Vertical distance = |4 – (–2)| = 6 units
34. Distance = |–3 – (–2)| = |–3 + 2| = 5 units? Wait step carefully: A(–2,–3), B(–2,2): vertical distance = |2 – (–3)| = 5 units ✔
35. Perimeter = 4 × side = 4 × 3 = 12 units
36. Coordinates = (5,4)
37. Check line: Slope between consecutive points = (4–3)/(2–1) = 1/1 =1; all slopes =1 → yes, straight line
38. Midpoint = ((–3+5)/2, (–4+2)/2) = (1, –1)
39. Reflection about X-axis: (x,y) → (x,–y); Example: (2,3) → (2,–3)
40. Area of triangle = ½ × base × height = ½ × 4 × 3 = 6 units²

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Section E: Application / Higher-Order Thinking – Answers

41. Rectangle vertices (–2,1), (2,1), (2,4), (–2,4)

• Length = 2 – (–2) = 4
• Width = 4 – 1 = 3
• Area = 4 × 3 = 12
• Perimeter = 2(4 + 3) = 14 units

42. Coordinates = (–3, –7), Quadrant III
43. Reflections:

• About Y-axis: (4, –3)
• About X-axis: (–4, 3)

44. Quadrants:

• (2,3) → I
• (–2,3) → II
• (–2,–3) → III
• (2,–3) → IV

45. Length = |4 – (–2)| = 6 units
46. Distance = |3 – (–3)| = 6 units
47. Midpoint = ((–4+2)/2, (6 + (–2))/2) = (–1, 2)
48. Point on X-axis 7 units right of origin: (7,0)
49. Rectangle area = (6–1) × (5–1) = 5 × 4 = 20 units²
50. Quadrants:

• (5,–3) → IV
• (–7,4) → II
• (–2,–6) → III
• (8,9) → I