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MathematicsClass 9Orienting Yourself: The Use of Coordinates

Orienting Yourself: The Use of Coordinates | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 5 min read

Orienting Yourself: The Use of Coordinates – this guide gives you a concise, exam-ready overview of Orienting Yourself: The Use of Coordinates from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

1.4 DISTANCE BETWEEN TWO POINTS IN THE 2-D PLANE

This section explains how to calculate the distance between any two points in the two-dimensional Cartesian plane. While distances between points lying on the same axis or forming line segments parallel to axes are straightforward (simply the absolute difference of their coordinates), the challenge arises when the segment joining two points is not parallel to either axis.

To solve this, the section applies the Baudhayana–Pythagoras theorem, which relates the lengths of sides in a right-angled triangle. By constructing a right triangle using the horizontal and vertical distances between the points as legs, the distance between the points becomes the hypotenuse.

For example, consider points A (3, 4) and D (7, 1). The horizontal distance (along the x-axis) is |7 - 3| = 4 units, and the vertical distance (along the y-axis) is |4 - 1| = 3 units. Using the theorem, the distance AD is √(4² + 3²) = 5 units.

This method generalizes to any two points (x₁, y₁) and (x₂, y₂), with the distance formula: Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

The section also discusses reflections of triangles across axes, showing that distances remain unchanged under reflection, illustrating the concept of congruence and symmetry in geometry.

Students are encouraged to think about how changes in coordinates affect distances and to apply the distance formula to various problems involving points in different quadrants.

📊 Diagram: Figure 1.6 shows triangle ADM in the first quadrant with points A (3,4), D (7,1), and M (9,6). Figure 1.7 illustrates the right triangle used to calculate distance AD with horizontal segment CD = 4 and vertical segment AC = 3. Figure 1.8 depicts the general distance formula between points (x₁, y₁) and (x₂, y₂). Figure 1.9 shows triangle ADM reflected across the y-axis with points A', M', D'.

🧪 Activity: Think and Reflect questions prompt students to analyze distances moved along axes and effects of reflection on distances.

🔗 Connection: Prepares students for end-of-chapter exercises involving plotting, distance calculations, and applications of coordinate geometry.

Frequently asked questions

Referring to Fig. 1.3, answer the following questions: (i) If \( \mathrm{D}_1\mathrm{R}_1 \) represents the door to Reiaan's room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis? (ii) What are the coordinates of \( \mathbf{D}_1 \) ? (iii) If \( \mathbf{R}_1 \) is the point (11.5, 0), how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily? (iv) If \( \mathrm{B}_1(0, 1.5) \) and \( \mathrm{B}_2(0, 4) \) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?

Solution:

(i) The door \( \mathrm{D}_1\mathrm{R}_1 \) is represented in Fig. 1.3. The distance from the left wall (y-axis) is the x-coordinate of the door points. Since \( \mathbf{D}_1 \) lies at some x-coordinate, the distance from the y-axis is that x-coordinate. Similarly, the distance from the x-axis is the y-coordinate of the door points.

(ii) The coordinates of \( \mathbf{D}_1 \) can be read from the figure. For example, if \( \mathbf{D}_1 \) lies at (11.5, 1.5), then the coordinates are

Think and Reflect 1. What are the standard widths for a room door? Look around your home and in school. 2. Are the doors in your school suitable for people in wheelchairs?

Answers will vary depending on observations:

1. Standard widths for room doors typically range from 0.75 meters to 1.0 meter. Wider doors (around 1.2 to 1.5 meters) are more comfortable and accessible.

2. Doors in schools may or may not be suitable for wheelchair users. Suitable doors should be at least 0.9 meters wide and have ramps or no steps. Students should observe and comment based on their school infrastructure.

Copy Fig. 1.4 and mark S and Q in your diagram. Mark any point P in Quadrant I and any point R in Quadrant III, and write down their coordinates.

Solution:

  • Point S is given as (3, -5), which lies in Quadrant IV.
  • Point Q is (-5, 3), which lies in Quadrant II.
  • Mark these points on a copy of Fig. 1.4.
  • Choose any point P in Quadrant I, for example (2, 3), where both coordinates are positive.
  • Choose any point R in Quadrant III, for example (-4, -2), where both coordinates are negative.
  • Write down the coordinates as above.
Think and Reflect 1. What is the x-coordinate of a point on the y-axis? 2. Is there a similar generalisation for a point on the x-axis? 3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer. 4. If $x \neq y$, then $(x, y) \neq (y, x)$; and $(x, y) = (y, x)$ if and only if $x = y$. Is this claim true?

Answers:

1. The x-coordinate of a point on the y-axis is always 0 because the point lies on the vertical axis.

2. Similarly, the y-coordinate of a point on the x-axis is always 0.

3. Point Q (y, x) coincides with point P (x, y) only if x = y. Otherwise, they are different points.

4. The claim is true. If x ≠ y, then (x, y) ≠ (y, x). If (x, y) = (y, x), then x must equal y.

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