# NCERT Class 10 Mathematics Chapter 7 Coordinate Geometry: Summary, Concepts, and Revision Notes | SwaVid

Coordinate Geometry is a branch of mathematics that uses a coordinate system to study geometry. It allows us to represent geometric figures using algebr...

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# Coordinate Geometry

## Concepts in this chapter

## The ideas this chapter keeps returning to

## Where this chapter usually trips people up

## Find the gaps before this chapter finds them for you.

## Related chapters

### Recap of Cartesian Coordinates

Chapter 7 · Class 10 Mathematics

Coordinate Geometry is a branch of mathematics that uses a coordinate system to study geometry. It allows us to represent geometric figures using algebraic equations and solve geometric problems using algebraic methods. This chapter builds upon your understanding from Class 9, introducing powerful formulas to calculate distances, find points that divide line segments, and determine the area of triangles, all using the coordinates of points.

Your study route

Key topics

This section revisits the basics of plotting points in a Cartesian plane, understanding the X and Y axes, the origin, and the four quadrants. Each point is uniquely identified by an ordered pair (x, y), where 'x' is the abscissa and 'y' is the ordinate.

Example

While no new example is explicitly given for this recap, the concept is foundational to all examples in the chapter, such as locating points like P(3,2) or Q(2,3) before calculating distance.

Watch out

Students often confuse the x-coordinate with the y-coordinate, or misinterpret the signs of coordinates in different quadrants, leading to incorrect plotting or formula application.

This section revisits the basics of plotting points in a Cartesian plane, understanding the X and Y axes, the origin, and the four quadrants. Each point is uniquely identified by an ordered pair (x, y), where 'x' is the abscissa and 'y' is the ordinate.

Tap the card for an example

Example

While no new example is explicitly given for this recap, the concept is foundational to all examples in the chapter, such as locating points like P(3,2) or Q(2,3) before calculating distance.

Why it matters

A strong understanding of the coordinate system is essential as it forms the basis for all subsequent formulas and applications in coordinate geometry. Incorrectly identifying coordinates leads to errors in calculations.

Watch out

Students often confuse the x-coordinate with the y-coordinate, or misinterpret the signs of coordinates in different quadrants, leading to incorrect plotting or formula application.

Ask at home

Ask your child to plot a few points (e.g., (2,3), (-1,4), (-3,-2), (4,-1)) on a graph paper and identify the quadrant each point lies in. Also, ask them to identify the abscissa and ordinate for each point.

This chapter introduces fundamental concepts of coordinate geometry, enabling the calculation of distances between points using the Distance Formula. Students learn to apply this formula to determine if points are collinear and to verify properties of various geometric figures like triangles and quadrilaterals. The Section Formula is then introduced to find the coordinates of a point that divides a line segment in a given ratio internally, with the Midpoint Formula being a special case. Finally, the chapter covers the Area of a Triangle formula, which allows for the calculation of a triangle's area given its vertices, and provides an alternative method to check for collinearity of three points. Mastery of these formulas is crucial for solving a wide range of geometric problems algebraically.

Chapter summary

This chapter introduces fundamental concepts of coordinate geometry, enabling the calculation of distances between points using the Distance Formula. Students learn to apply this formula to determine if points are collinear and to verify properties of various geometric figures like triangles and quadrilaterals. The Section Formula is then introduced to find the coordinates of a point that divides a line segment in a given ratio internally, with the Midpoint Formula being a special case. Finally, the chapter covers the Area of a Triangle formula, which allows for the calculation of a triangle's area given its vertices, and provides an alternative method to check for collinearity of three points. Mastery of these formulas is crucial for solving a wide range of geometric problems algebraically.

What you should learn

Keep these close

The Cartesian plane uses two perpendicular axes (X and Y) to locate points, with the origin at (0,0).

A point's coordinates are written as (x, y), where x is the abscissa and y is the ordinate.

The distance formula between P(x1, y1) and Q(x2, y2) is d = sqrt((x2-x1)^2 + (y2-y1)^2).

Three points A, B, C are collinear if AB + BC = AC (or any similar permutation of distances).

The section formula for a point P(x,y) dividing a line segment joining A(x1,y1) and B(x2,y2) internally in the ratio m1:m2 is x = (m1x2 + m2x1)/(m1+m2) and y = (m1y2 + m2y1)/(m1+m2).

The midpoint formula is a special case of the section formula (ratio 1:1): x = (x1+x2)/2 and y = (y1+y2)/2.

The area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|.

If the area of a triangle formed by three points is zero, then the points are collinear.

Properties of geometric figures (e.g., square, rhombus, equilateral triangle) can be verified using the distance formula by comparing side lengths and diagonal lengths.

The distance of a point (x,y) from the origin (0,0) is sqrt(x^2 + y^2).

Always pay attention to the signs of coordinates and the order of operations when applying formulas.

Area is always a positive quantity; the absolute value in the area formula ensures this.

Common confusions

It is easy to think

Forgetting to take the square root in the distance formula, often leaving the answer as (distance)^2.

The clearer idea

The formula d^2 = (x2-x1)^2 + (y2-y1)^2 gives the square of the distance. The final step is always to take the positive square root to find the actual distance 'd'.

It is easy to think

Confusing the order of coordinates (x,y) or mixing up x1, x2, y1, y2 in formulas, especially the section formula.

The clearer idea

Always remember that coordinates are (x-value, y-value). For formulas involving two points, consistently assign (x1, y1) to the first point and (x2, y2) to the second. For the section formula, ensure m1 is multiplied by x2/y2 and m2 by x1/y1.

It is easy to think

Believing that a negative value for the area of a triangle is possible, or not using the absolute value.

The clearer idea

Area is a measure of space and is always a non-negative quantity. The absolute value in the area formula ensures the result is positive. If the calculation yields a negative number, take its absolute value. A zero area indicates collinear points.

It is easy to think

Assuming that the section formula can be used for external division, which is not part of the Class 10 NCERT syllabus.

The clearer idea

The NCERT Class 10 syllabus for the section formula exclusively covers internal division. The formula provided in the textbook and used in examples is specifically for a point lying *between* the two given points.

Before you push ahead

Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns.

Keep exploring Class 10

Source

- Calculate the distance between any two given points in a Cartesian plane.
- Determine if three given points are collinear using appropriate formulas (distance or area).
- Apply the distance formula to verify properties of triangles (e.g., equilateral, isosceles, right-angled) and quadrilaterals (e.g., square, rhombus, parallelogram).
- Find the coordinates of a point that divides a line segment in a given ratio internally using the section formula.
- Calculate the area of a triangle when the coordinates of its vertices are given.
- Calculate the distance between any two given points in a Cartesian plane.
- Determine if three given points are collinear using appropriate formulas (distance or area).
- Apply the distance formula to verify properties of triangles (e.g., equilateral, isosceles, right-angled) and quadrilaterals (e.g., square, rhombus, parallelogram).
- Find the coordinates of a point that divides a line segment in a given ratio internally using the section formula.
- Calculate the area of a triangle when the coordinates of its vertices are given.
- NCERT Class 10 Mathematics textbook: Mathematics : Chapter 7: Coordinate Geometry

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