What is the purpose of the section formula?

The coordinates of a point splitting a line segment in a specified ratio can be found using the section formula.

In what way can I determine whether the three points are collinear?

The area formula can be used to determine whether three points are collinear if the area of the triangle they create is zero.

Why is it important to solve Coordinate Geometry problems from the coordinate geometry class 10 MCQ?

Exam problem-solving abilities are strengthened, and a greater comprehension of the question structure is gained by completing the coordinate geometry class 10 MCQ along with the previous year's questions.

In coordinate geometry, how do you locate the section where a point is located?

You can identify the quadrant in which the point dividing a line segment in a certain ratio is located by applying the section formula.

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Coordinate Geometry

1.0Master Cartesian Planes, Distance Rules, and Section Division in Minutes

Unlock the algebraic system that maps geometric space. Learn how to plot positions across four distinct quadrants, master the Pythagorean basis of the distance formula, split line segments precisely using internal ratio partitions, and compute spatial areas to ace your Class 10 board exams.

Class: 10 Mathematics (CBSE)
Chapter: Coordinate Geometry
Estimated Learning Time: 20–25 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Identify and describe components of the Cartesian Plane, including the abscissa, ordinate, and four quadrants.
Apply the Distance Formula to compute the lengths of line segments between any two coordinate pairs.
Implement the Section Formula to find coordinates that partition a line segment in a specified ratio ($m : n$).
Deduce the coordinates of a line segment's midpoint.
Calculate the area of a triangle given three coordinate vertices.
Verify the Collinearity of three coordinates by evaluating if their enclosed triangular area equals zero.

3.0Introduction to Coordinate Geometry

Welcome to the mathematical bridge where geometry meets algebra! Coordinate Geometry is a branch of mathematics where spatial figures, lines, and curves are analyzed analytically using algebraic equations. In this lesson, we break down the two-dimensional Cartesian Plane created by the intersection of perpendicular horizontal and vertical axes. You will learn to easily locate ordered numeric pairs $(x, y)$, calculate the straight-line distance between any two coordinates using the Distance Formula, and compute exact internal partition points along a line segment using the Section Formula. Finally, you will learn how to calculate the area enclosed by three points and apply this tool to mathematically prove whether points lie along a single straight line (collinearity).

4.0

Coordinate Geometry is the branch of mathematics where the relation between geometric shapes and algebraic equations (linear equations and quadratic equations) is studied in terms of a coordinate. In Class 10th Maths Coordinate Geometry is used to understand how geometric concepts relate to algebra.

Different quadrants in coordinate geometry

5.0The Cartesian Plane

Coordinate Geometry is used to visually represent the position of a given coordinate in a two-dimensional plane, also known as a cartesian plane. The Cartesian plane is a system of two perpendicular axes intersecting each other at 90. The horizontal or x-axis is known as abscissa, and the vertical or y-axis is called the ordinate.

The abscissa and ordinate intersect at the origin (0,0), dividing the plane into four quadrants, namely:

Quadrant I: Both x & y are positive.
Quadrant II: x is negative, y is positive.
Quadrant III: Both x & y are negative.
Quadrant IV: x is positive, y is negative.

6.0Important Concepts for Class 10 Coordinate Geometry

Coordinates of a Point:

In coordinate geometry, a point can be represented as an ordered pair (x, y), also known as coordinates, which indicates its position on the Cartesian plane. These coordinates are crucial for understanding how geometric shapes or points are located and measured in the plane.

Distance Formula:

The distance formula is an important concept of coordinate geometry, used to find straight line distance between two points P(x₁,y₁) & Q(x₂,y₂), as shown in the figure.

The distance(d) may be calculated with the formula, expressed as:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

This formula for determining the distance between two coordinates is the dominating formula for many coordinate geometry class 10 NCERT solutions.

Section Formula (Internal Division):

The section formula is used to find the coordinates of a point dividing a given line segment in a certain ratio. If a point, say, P(x, y) divides the line segment joining two points A(x₁, y₁) & B(x₂, y₂) in the ratio m : n, the coordinates of point P are given by:

$x = \frac{n x _{1} + m x _{2}}{m + n}, y = \frac{n y _{1} + m y _{2}}{m + n}$

For midpoints where the ratio will be 1:1 then the formula is:

$x = \frac{x _{1} + x _{2}}{2}, y = \frac{y _{1} + y _{2}}{2}$

The above-mentioned formula is used to solve a wide range of coordinate geometry class 10 important questions, which include finding the ratio dividing the line segments.

Area of a Triangle:

The area of a triangle which is formed by any three coordinates, say, A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), in coordinate geometry can be found with the help of the following formula:

$Area = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$

formula for area of a triangle in coordinate geometry

In coordinate geometry class 10 solutions, you will often come across problems where you need to find the collinearity of any 3 coordinates. The formula mentioned above is used to find the collinearity of the given three points. Collinearly is in the same straight line. The area of the triangle will be 0 if the points are collinear.

Area of triangle = 0; if the points are collinear

7.0Problems On Class 10 Coordinate Geometry

Problem: Find the area of the triangle ABC with A (1, –4) and the mid-points of sides through A being (2, – 1) and (0, – 1).

Solution: Let the other coordinates of the triangle be B (a,b) and C(x,y). Let the midpoints of the triangle be D (2, – 1) and E(0, – 1)

In the side AB,

$2 = \frac{1 + a}{2}$ , $-1 = \frac{- 4 + b}{2}$

$4 = 1 + a$ $-2 = -4 + b$

$a = 3$ $b = 2$

Coordinates of B (3,-3)

In the side AC,

$0 = \frac{1 + x}{2}$ , $-1 = \frac{- 4 + b}{2}$

$0 = 1 + a$ $-2 = -4 + b$

$a = -1$ $b = 2$

Coordinates of C (-1,2)

$Area of triangle ABC = \frac{1}{2} [1 (2 - 2) + 3 (2 + 4) -1 (- 4 - 2)]$

$= \frac{1}{2} [18 + 6]$

= 12 square units.

Problem 2: The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1 : 2 internally lies in which section.

Solution: Coordinates the point dividing the line segment (7, –6) and (3, 4). Let n = 1 and m = 2

$x = \frac{n x _{1} + m x _{2}}{m + n}, y = \frac{n y _{1} + m y _{2}}{m + n}$

$x = \frac{2 ( 7 ) + 1 ( 3 )}{3}, y = \frac{1 ( - 6 ) + 2 ( 4 )}{3}$

$x = 17/3, y = 2/3$

Both the points are positive. Hence, the coordinates lie in the I quadrant.

Problem 3: What will be the distance between points A (0, 6) and B (0, –2)?

Solution: Using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

$d = (0 - 0)^{2} + (6 - (- 2))^{2}$

$d = 8^{2} = 8 u ni t s$

8.0Important topics in Class 10 Maths: Coordinate Geometry

Magnetic fields

Electric motor

9.0EUREKA by ALLEN – Learn Smarter for Class 10 Exams

With EUREKA from the ALLEN, you will achieve the highest level of academic achievement through an innovative approach designed specifically for Class 10 students. EUREKA provides world-class education using top instructors, cutting-edge artificial intelligence, and a proven method for preparing for board examinations; thus creating a complete learning environment for success. Through personalized assistance, compelling content, and ongoing assessment of progress toward maximum achievement, students receive everything they need to realise their full academic capabilities.

Interactive concept-based learning Story-driven video lessons Board exam-oriented preparation Subjective answer writing practice Instant quizzes and performance feedback Real-time progress monitoring AI-powered doubt support available 24×7 NCERT & CBSE-aligned curriculum Flexible on-demand learning access

Explore Now

10.0Supporting Study Materials

This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 7 of Class 10 Maths, follows the latest NCERT guidelines. Complete with quadrant mapping grids, right-angled distance derivations, and section division layout blocks, this guide provides thorough preparation for your board examinations.

CBSE Class 10 Maths Notes Chapter 7 Coordinate Geometry	NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry

30-Second Quick Review: Coordinate Geometry

Every point is represented as (x, y).
The x-coordinate is called the abscissa.
The y-coordinate is called the ordinate.
Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Midpoint Formula:

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

Distance is always positive.
Equal distances help determine collinearity.
Coordinate Geometry combines Algebra and Geometry to solve practical problems.

11.0Previous Year Questions (PYQs) on Coordinate Geometry

Question: Find the value of x if the distance between the points (x, 3) and (5, 3) is 8 units.

Answer: Since the y-coordinates are equal,

Distance = |x – 5|

|x – 5| = 8

Therefore,

x = 13 or x = –3

Question: Determine whether the points A(1, 2), B(4, 6), and C(7, 10) are collinear.

Answer: Using the distance formula:

AB = 5

BC = 5

AC = 10

Since,

AB + BC = AC

Therefore,

The three points are collinear.

12.0Recommended Next Topics

Real Numbers
Polynomials
Pair of Linear Equations in Two Variables
Quadratic Equations

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Coordinate Geometry

1.0Master Cartesian Planes, Distance Rules, and Section Division in Minutes

Unlock the algebraic system that maps geometric space. Learn how to plot positions across four distinct quadrants, master the Pythagorean basis of the distance formula, split line segments precisely using internal ratio partitions, and compute spatial areas to ace your Class 10 board exams.

Class: 10 Mathematics (CBSE)
Chapter: Coordinate Geometry
Estimated Learning Time: 20–25 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Identify and describe components of the Cartesian Plane, including the abscissa, ordinate, and four quadrants.
Apply the Distance Formula to compute the lengths of line segments between any two coordinate pairs.
Implement the Section Formula to find coordinates that partition a line segment in a specified ratio ($m : n$).
Deduce the coordinates of a line segment's midpoint.
Calculate the area of a triangle given three coordinate vertices.
Verify the Collinearity of three coordinates by evaluating if their enclosed triangular area equals zero.

3.0Introduction to Coordinate Geometry

Welcome to the mathematical bridge where geometry meets algebra! Coordinate Geometry is a branch of mathematics where spatial figures, lines, and curves are analyzed analytically using algebraic equations. In this lesson, we break down the two-dimensional Cartesian Plane created by the intersection of perpendicular horizontal and vertical axes. You will learn to easily locate ordered numeric pairs $(x, y)$, calculate the straight-line distance between any two coordinates using the Distance Formula, and compute exact internal partition points along a line segment using the Section Formula. Finally, you will learn how to calculate the area enclosed by three points and apply this tool to mathematically prove whether points lie along a single straight line (collinearity).

4.0

Coordinate Geometry is the branch of mathematics where the relation between geometric shapes and algebraic equations (linear equations and quadratic equations) is studied in terms of a coordinate. In Class 10th Maths Coordinate Geometry is used to understand how geometric concepts relate to algebra.

5.0The Cartesian Plane

Coordinate Geometry is used to visually represent the position of a given coordinate in a two-dimensional plane, also known as a cartesian plane. The Cartesian plane is a system of two perpendicular axes intersecting each other at 90. The horizontal or x-axis is known as abscissa, and the vertical or y-axis is called the ordinate.

The abscissa and ordinate intersect at the origin (0,0), dividing the plane into four quadrants, namely:

Quadrant I: Both x & y are positive.
Quadrant II: x is negative, y is positive.
Quadrant III: Both x & y are negative.
Quadrant IV: x is positive, y is negative.

6.0Important Concepts for Class 10 Coordinate Geometry

Coordinates of a Point:

In coordinate geometry, a point can be represented as an ordered pair (x, y), also known as coordinates, which indicates its position on the Cartesian plane. These coordinates are crucial for understanding how geometric shapes or points are located and measured in the plane.

Distance Formula:

The distance formula is an important concept of coordinate geometry, used to find straight line distance between two points P(x₁,y₁) & Q(x₂,y₂), as shown in the figure.

The distance(d) may be calculated with the formula, expressed as:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

This formula for determining the distance between two coordinates is the dominating formula for many coordinate geometry class 10 NCERT solutions.

Section Formula (Internal Division):

The section formula is used to find the coordinates of a point dividing a given line segment in a certain ratio. If a point, say, P(x, y) divides the line segment joining two points A(x₁, y₁) & B(x₂, y₂) in the ratio m : n, the coordinates of point P are given by:

$x = \frac{n x _{1} + m x _{2}}{m + n}, y = \frac{n y _{1} + m y _{2}}{m + n}$

For midpoints where the ratio will be 1:1 then the formula is:

$x = \frac{x _{1} + x _{2}}{2}, y = \frac{y _{1} + y _{2}}{2}$

The above-mentioned formula is used to solve a wide range of coordinate geometry class 10 important questions, which include finding the ratio dividing the line segments.

Area of a Triangle:

The area of a triangle which is formed by any three coordinates, say, A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), in coordinate geometry can be found with the help of the following formula:

$Area = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$

In coordinate geometry class 10 solutions, you will often come across problems where you need to find the collinearity of any 3 coordinates. The formula mentioned above is used to find the collinearity of the given three points. Collinearly is in the same straight line. The area of the triangle will be 0 if the points are collinear.

Area of triangle = 0; if the points are collinear

7.0Problems On Class 10 Coordinate Geometry

Problem: Find the area of the triangle ABC with A (1, –4) and the mid-points of sides through A being (2, – 1) and (0, – 1).

Solution: Let the other coordinates of the triangle be B (a,b) and C(x,y). Let the midpoints of the triangle be D (2, – 1) and E(0, – 1)

In the side AB,

$2 = \frac{1 + a}{2}$ , $-1 = \frac{- 4 + b}{2}$

$4 = 1 + a$ $-2 = -4 + b$

$a = 3$ $b = 2$

Coordinates of B (3,-3)

In the side AC,

$0 = \frac{1 + x}{2}$ , $-1 = \frac{- 4 + b}{2}$

$0 = 1 + a$ $-2 = -4 + b$

$a = -1$ $b = 2$

Coordinates of C (-1,2)

$Area of triangle ABC = \frac{1}{2} [1 (2 - 2) + 3 (2 + 4) -1 (- 4 - 2)]$

$= \frac{1}{2} [18 + 6]$

= 12 square units.

Problem 2: The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1 : 2 internally lies in which section.

Solution: Coordinates the point dividing the line segment (7, –6) and (3, 4). Let n = 1 and m = 2

$x = \frac{n x _{1} + m x _{2}}{m + n}, y = \frac{n y _{1} + m y _{2}}{m + n}$

$x = \frac{2 ( 7 ) + 1 ( 3 )}{3}, y = \frac{1 ( - 6 ) + 2 ( 4 )}{3}$

$x = 17/3, y = 2/3$

Both the points are positive. Hence, the coordinates lie in the I quadrant.

Problem 3: What will be the distance between points A (0, 6) and B (0, –2)?

Solution: Using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

$d = (0 - 0)^{2} + (6 - (- 2))^{2}$

$d = 8^{2} = 8 u ni t s$

8.0Important topics in Class 10 Maths: Coordinate Geometry

Magnetic fields

Electric motor

9.0EUREKA by ALLEN – Learn Smarter for Class 10 Exams

With EUREKA from the ALLEN, you will achieve the highest level of academic achievement through an innovative approach designed specifically for Class 10 students. EUREKA provides world-class education using top instructors, cutting-edge artificial intelligence, and a proven method for preparing for board examinations; thus creating a complete learning environment for success. Through personalized assistance, compelling content, and ongoing assessment of progress toward maximum achievement, students receive everything they need to realise their full academic capabilities.

Interactive concept-based learning Story-driven video lessons Board exam-oriented preparation Subjective answer writing practice Instant quizzes and performance feedback Real-time progress monitoring AI-powered doubt support available 24×7 NCERT & CBSE-aligned curriculum Flexible on-demand learning access

Explore Now

10.0Supporting Study Materials

This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 7 of Class 10 Maths, follows the latest NCERT guidelines. Complete with quadrant mapping grids, right-angled distance derivations, and section division layout blocks, this guide provides thorough preparation for your board examinations.

CBSE Class 10 Maths Notes Chapter 7 Coordinate Geometry	NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry

30-Second Quick Review: Coordinate Geometry

Every point is represented as (x, y).
The x-coordinate is called the abscissa.
The y-coordinate is called the ordinate.
Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Midpoint Formula:

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

Distance is always positive.
Equal distances help determine collinearity.
Coordinate Geometry combines Algebra and Geometry to solve practical problems.

11.0Previous Year Questions (PYQs) on Coordinate Geometry

Question: Find the value of x if the distance between the points (x, 3) and (5, 3) is 8 units.

Answer: Since the y-coordinates are equal,

Distance = |x – 5|

|x – 5| = 8

Therefore,

x = 13 or x = –3

Question: Determine whether the points A(1, 2), B(4, 6), and C(7, 10) are collinear.

Answer: Using the distance formula:

AB = 5

BC = 5

AC = 10

Since,

AB + BC = AC

Therefore,

The three points are collinear.

12.0Recommended Next Topics

Real Numbers
Polynomials
Pair of Linear Equations in Two Variables
Quadratic Equations

Frequently Asked Questions

What is the purpose of the section formula?

In what way can I determine whether the three points are collinear?

Why is it important to solve Coordinate Geometry problems from the coordinate geometry class 10 MCQ?

In coordinate geometry, how do you locate the section where a point is located?

Frequently Asked Questions

What is the purpose of the section formula?

In what way can I determine whether the three points are collinear?

Why is it important to solve Coordinate Geometry problems from the coordinate geometry class 10 MCQ?

In coordinate geometry, how do you locate the section where a point is located?

Join ALLEN!

Coordinate Geometry

1.0Master Cartesian Planes, Distance Rules, and Section Division in Minutes

2.0Learning Outcomes

3.0Introduction to Coordinate Geometry

4.0

5.0The Cartesian Plane

6.0Important Concepts for Class 10 Coordinate Geometry

7.0Problems On Class 10 Coordinate Geometry

8.0Important topics in Class 10 Maths: Coordinate Geometry

9.0EUREKA by ALLEN – Learn Smarter for Class 10 Exams

10.0Supporting Study Materials

30-Second Quick Review: Coordinate Geometry

11.0Previous Year Questions (PYQs) on Coordinate Geometry

12.0Recommended Next Topics

On this page

Join ALLEN!

Coordinate Geometry

1.0Master Cartesian Planes, Distance Rules, and Section Division in Minutes

2.0Learning Outcomes

3.0Introduction to Coordinate Geometry

4.0

5.0The Cartesian Plane

6.0Important Concepts for Class 10 Coordinate Geometry

7.0Problems On Class 10 Coordinate Geometry

8.0Important topics in Class 10 Maths: Coordinate Geometry

9.0EUREKA by ALLEN – Learn Smarter for Class 10 Exams

10.0Supporting Study Materials

30-Second Quick Review: Coordinate Geometry

11.0Previous Year Questions (PYQs) on Coordinate Geometry

12.0Recommended Next Topics

On this page