CBSE 10 Coordinate Geometry: Plotting Your Path to Success in the Two-Dimensional Plane (Formulas Included)

We know that coordinate geometry is an important topic for you. In this article, we will explore two-dimensional coordinate geometry, an essential tool for describing the position of points in a plane.

Coordinate geometry in two dimensional plane

In coordinate geometry, we use a pair of numbers called coordinates to represent the position of a point in a plane. The two numbers are usually denoted by (x,y), where x is the horizontal coordinate and y is the vertical coordinate. The point where the two axes intersect is called the origin, and it has coordinates (0,0).

Distance between two points

The distance between two points in a plane can be found using the distance formula:

d = sqrt[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁,y₁) and (x₂,y₂) are the coordinates of the two points.

Reflection of a point across the x-axis

To reflect a point across the x-axis, we simply change the sign of the y-coordinate. For example, if we have a point (x,y), its reflection across the x-axis is (x,-y).

Reflection of a point across the y-axis

To reflect a point across the y-axis, we simply change the sign of the x-coordinate. For example, if we have a point (x,y), its reflection across the Y-axis is (-x,y).

Section formula

The section formula is used to find the coordinates of a point that lies on a line segment joining two given points. If we have two points A(x₁,y₁) and B(x₂,y₂), and we want to find the coordinates of a point P that divides the line segment AB in the ratio m:n, then the coordinates of P are given by:

P = ((mx₂ + nx₁) / (m + n),(my₂ + ny₁) / (m + n))

Solved examples

Example 1:

Find the distance between the points A(2,3) and B(5,7).

✅  Solution: d = sqrt[(5 - 2)² + (7 - 3)²] = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.

Example 2:

Find the point P that divides the line segment joining A(-3,2) and B(7,8) in the ratio 2:3.

✅  Solution:
      1. Let the coordinates of P be (x,y).
      2. Then x = [2 * 7 + 3 * (-3)] / (2 + 3) = 1.
      3. Similarly, y = (2 * 8 + 3 * 2) / (2 + 3) = 4.4
      4. Therefore, the coordinates of P are (1,4.4).

FAQs

1. What is coordinate geometry used for?

Coordinate geometry is used to describe the positions of points in a plane, and it is an essential tool in many fields, including mathematics, physics, and engineering.

2. What is the distance formula in coordinate geometry?

The distance formula in coordinate geometry is:

d = sqrt[(x₂ - x₁)² + (y₂ - y₁)²]

where (x₁,y₁) and (x₂,y₂) are the coordinates of the two points.

3. How do you reflect a point across the x-axis?

To reflect a point across the x-axis, you simply change the sign of the y-coordinate.

4. How is the section formula used in coordinate geometry?

The section formula is used to find the coordinates of a point that lies on a line segment joining two given points. It can be used to solve problems involving dividing a line segment in a given ratio.

Conclusion

Coordinate geometry is a fascinating and important topic that allows us to describe the positions of points in a plane. By understanding the concepts we have covered in this article, you will be well-equipped to tackle more advanced problems in coordinate geometry. For more resources on mathematics and other subjects, visit Aha AI.

Relevant links

• CBSE 10 Geometry: Unravel the Enigma of Triangles (Solved Examples)
• NCERT Solutions for Class 10 Mathematics - Coordinate Geometry
• Two-Dimensional Coordinate Geometry by BYJU'S
• Two-Dimensional Coordinate Geometry by Vendantu