Coordinate Geometry Class 10 ||Maths|| Chapter 7 NCERT Notes

Coordinate Geometry Class 10 ||Maths|| Chapter 7 NCERT Notes

Introduction to the Coordinate Plane:

The coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular lines:

  • The X-axis (horizontal line).
  • The Y-axis (vertical line).

These axes divide the plane into four quadrants:

  1. Quadrant I: Both x and y are positive.
  2. Quadrant II: x is negative, y is positive.
  3. Quadrant III: Both x and y are negative.
  4. Quadrant IV: x is positive, y is negative.

Each point in this plane is defined by an ordered pair (x, y), called coordinates. The first number is the x-coordinate (or abscissa), and the second number is the y-coordinate (or ordinate).

Important Formulas in Coordinate Geometry:

1. Distance Formula:

The distance between two points (x1,y1) and (x2,y2) in a coordinate plane is given by the formula:

Distance=(x2x1)2+(y2y1)2
  • This formula is derived from the Pythagoras Theorem in a right-angled triangle.

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

Using the formula:

Distance=(52)2+(73)2=9+16=25=5

2. Section Formula:

The section formula helps us find the coordinates of a point that divides a line segment joining two points in a given ratio.

  • For internal division: If the point P(x,y) divides the line segment joining two points (x1,y1) and (x2,y2) in the ratio m1:m2, then the coordinates of P are:
P(x,y)=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)
  • For external division: The formula remains the same, but the sign between m1 and m2 will change.

Application Example: Find the point that divides the line segment joining points A(2,3) and B(6,1) in the ratio 2:1 internally.

Using the section formula:

P(x,y)=(2(6)+1(2)2+1,2(1)+1(3)2+1)P(x,y)=(12+23,233)=(143,13)

Thus, P(143,13) is the required point.

3. Midpoint Formula:

The midpoint of a line segment joining two points (x1,y1) and (x2,y2) can be found using the midpoint formula:

MidpointM(x,y)=(x1+x22,y1+y22)

Application Example: Find the midpoint of the line segment joining A(1,4) and B(5,6).

Using the formula:

M(x,y)=(1+52,4+62)=(3,5)

Thus, the midpoint is M(3,5).

4. Area of a Triangle:

The area of a triangle with vertices at points (x1,y1)(x2,y2), and (x3,y3) is given by the formula:

Area of triangle=12x1(y2y3)+x2(y3y1)+x3(y1y2)
  • The absolute value ensures that the area is positive, regardless of the order of the points.

Application Example: Find the area of the triangle with vertices A(2,3)B(4,7), and C(6,3).

Using the formula:

Area=122(73)+4(33)+6(37)=122(4)+4(0)+6(4)=128+024=12×16=8square units

Thus, the area of the triangle is 8 square units.

Important Concepts to Remember:

  • Collinearity of Three Points:
    If the area of a triangle formed by three points is zero, then the points are said to be collinear, i.e., they lie on the same straight line.

  • Slope of a Line:
    Though not a part of the main chapter in Class 10, it's helpful to know that the slope of a line joining two points (x1,y1) and (x2,y2) is:

    Slope (m)=y2y1x2x1
  • Quadrants Overview:
    Each quadrant is determined by the sign of x and y coordinates:

    • Quadrant I: (+, +)
    • Quadrant II: (-, +)
    • Quadrant III: (-, -)
    • Quadrant IV: (+, -)

Key Takeaways:

  1. Distance Formula helps to find the distance between two points in a plane.
  2. Section Formula and Midpoint Formula help in dividing a line segment in a given ratio or finding the midpoint of the segment.
  3. Area of a Triangle Formula is useful to determine the area when the vertices are known in coordinate form.
  4. Quick-thinking Tip: If you ever need to check the collinearity of points, calculate the area of the triangle. If it’s zero, they are collinear.