Coordinate Geometry is a chapter in the CBSE Class 10 Mathematics syllabus from Mathematics. This chapter hub brings together revision notes, practice questions, worksheets, flashcards, formula sheet to help students learn, practice, and revise Coordinate Geometry effectively.

Scroll down to find Coordinate Geometry notes, practice questions, worksheets, and revision resources — all in one place. Use the sidebar to jump to any section, or browse the full page below.

Coordinate Geometry

NCERT Class 10 Mathematics Chapter 7: Coordinate Geometry (Pages 99–112)

Class 10 CBSE hub Mathematics chapters Mathematics

Summary of Coordinate Geometry

Playing 00:00 / 00:00

Coordinate Geometry at a Glance

Board

CBSE

Class

Class 10

Subject

Mathematics

Book

Mathematics

Chapter

Pages

99–112

Resources

7 study resources

Coordinate Geometry Summary

In this chapter, students will explore coordinate geometry, which combines algebra and geometry to analyze positions on a plane. The foundation of this subject lies in understanding coordinate axes, where the horizontal line represents the x-axis and the vertical line represents the y-axis. Each point on this plane corresponds to a pair of numbers called coordinates. The first number indicates the distance from the y-axis, while the second number indicates the distance from the x-axis. This relationship is crucial for understanding how to plot points and draw shapes accurately. Next, we discuss the distance between two points on the coordinate plane. To find this distance, students learn to apply the Pythagorean theorem. For example, if two points lie on the x-axis, the distance is the difference between their x-coordinates. This principle holds true for points on the y-axis as well. The knowledge extends to finding the distance between points not aligned with the axes by visualizing them in a right triangle formation. The hypotenuse of this triangle represents the distance between the two points and can be calculated using the Pythagorean theorem. This leads to the formulation of the distance formula, a key tool in coordinate geometry that allows for finding distances between any two points in the plane. Students will also learn about the section formula, which helps in finding a point that divides a line segment into a specific ratio. This concept is especially useful for applications in real-world scenarios, such as determining the location of objects based on their relative positions. The chapter will feature examples to reinforce these ideas, such as verifying whether three points form a triangle and finding relationships among points based on their distances. Overall, this chapter lays the groundwork for students to use coordinate geometry practically, linking mathematical theories with everyday applications in fields like physics, engineering, and navigation. Understanding coordinate geometry is essential for developing spatial awareness and enhancing problem-solving skills, making mathematics more accessible and enjoyable.

Coordinate Geometry Revision Guide

Download the Coordinate Geometry revision guide with key points, summaries, and quick revision notes for CBSE Class 10 Mathematics.

Key Points

Coordinate System Basics.

The coordinate plane consists of x-axis (horizontal) and y-axis (vertical), intersecting at the origin (0, 0). Points are represented as (x, y) where x is the abscissa and y is the ordinate.

Distance Formula Definition.

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by: PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]. This formula derives from the Pythagorean theorem.

Distance from Origin.

Distance of point P(x, y) from origin O(0, 0) is given by OP = √(x² + y²). It's essential for quickly evaluating distances in geometry.

Midpoint Formula.

The midpoint M of line segment AB joining A(x₁, y₁) and B(x₂, y₂) is: M = [(x₁ + x₂)/2, (y₁ + y₂)/2]. This provides the center point between two coordinates.

Section Formula.

To find a point P that divides the segment AB in the ratio m:n, use P = [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)]. This is crucial for segment division problems.

Collinearity Criterion.

Points A, B, and C are collinear if the area of triangle ABC is zero, which happens when: (y₂ - y₁)(x₃ - x₁) = (y₃ - y₁)(x₂ - x₁).

Slope of a Line.

The slope (m) of line joining points (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁) / (x₂ - x₁). It indicates the steepness of the line.

Parallel Lines.

Two lines are parallel if they have the same slope. This property is used frequently in geometry ideas involving multiple lines.

Perpendicular Lines.

Two lines are perpendicular if the product of their slopes is -1. If one line has slope m, the other will be -1/m.

Equation of a Line.

A line in slope-intercept form is expressed as y = mx + b where m is the slope and b is the y-intercept.

Conic Sections Overview.

Conic sections (circles, parabolas, ellipses, and hyperbolas) can be described algebraically using coordinate geometry.

Applications in Real Life.

Coordinate geometry is used in various fields such as navigation, physics, and engineering to solve practical problems.

Identifying Triangle Types.

Use coordinate distances between points to classify triangles as scalene, isosceles, or equilateral based on length comparisons.

Finding Area of Triangles.

Triangle area can be calculated using coordinates via formula: Area = 0.5 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.

Using Locus Concepts.

The set of points satisfying a particular condition (locus property) can be analyzed in coordinate geometry, such as circles being equidistant from a center.

Real-World Graphing.

Plots of functions or data points visually represent relationships, assisting in predicting trends or outcomes.

Common Misconceptions.

Students often confuse distance from origin with coordinates or miscalculate distances; always apply the distance formula correctly.

Set Up Axes Correctly.

Carefully draw the x and y axes perpendicularly. Mistakes here lead to miscalculated distances.

Practice with Graphs.

Frequent practice plotting points and lines will improve understanding of their relationships in a coordinate system.

Exam Tip: Diagrams!

In exams, sketching diagrams for coordinate problems can clarify calculations and improve accuracy on answers.

Coordinate Geometry Practice Questions & Answers

Practice important questions and exam-style problems from Coordinate Geometry. These questions cover key topics from the CBSE Class 10 Mathematics syllabus.

How to practice: Start with the questions below to test your understanding of Coordinate Geometry. Use the revision guide to review concepts you find difficult, then come back and retry the questions for better retention.

What are the coordinates of a point located on the x-axis?

Single Answer MCQ

Q-00174020

View explanation

In the coordinate plane, the point (0, y) lies on which axis?

Single Answer MCQ

Q-00174021

View explanation

If the coordinates of a point are (3, -4), what is its distance from the x-axis?

Single Answer MCQ

Q-00174022

View explanation

What is the formula to find the distance between two points (x1, y1) and (x2, y2)?

Single Answer MCQ

Q-00174023

View explanation

The point (4, 5) is located in which quadrant of the Cartesian plane?

Single Answer MCQ

Q-00174024

View explanation

Which of the following represents a linear equation in two variables?

Single Answer MCQ

Q-00174025

View explanation

Which of the following points lies on the line described by the equation y = 2x + 1?

Single Answer MCQ

Q-00174026

View explanation

If the coordinates (x, y) divide the line segment between (2, 3) and (8, 7) in the ratio 2:3, what are the coordinates of (x, y)?

Single Answer MCQ

Q-00174027

View explanation

View all 102 Coordinate Geometry questions

What is the distance between the points (1, 2) and (4, 6)?

What is the y-intercept of the line represented by the equation 3x + 2y = 6?

What are the coordinates of a point that divides the line segment joining A(2, 3) and B(4, 7) in the ratio 1:3?

Which point is equidistant from the points (0, 0) and (6, 8)?

Find the coordinates of the point that divides the line segment between A(1, 4) and B(7, 10) in the ratio 3:5.

Which of the following is a correct description of the point (x, y) when x is negative and y is positive?

What is the ratio in which the point (3, 5) divides the line segment joining A(1, 1) and B(5, 9)?

Which coordinate pair would represent a point 5 units left and 3 units up from (0, 0)?

If point P divides A(4, 2) and B(2, 6) in the ratio 2:1, what are the coordinates of P?

If A(1,3), B(1,7), and C(7,3) are vertices of a triangle, how can you find the area of this triangle?

For points A(0, 0) and B(10, 10), what point divides the segment in the ratio 4:6?

What is the slope of the line that passes through points (2, 3) and (5, 11)?

What are the coordinates of point P that divides A(2, -3) and B(-6, 5) in the ratio 3:1?

In a two-dimensional plane, what geometric representation corresponds to the equation x + y = 5?

A point divides the segment joining A(3, 4) and B(-3, -4) in the ratio 1:2. What are its coordinates?

In what ratio does the line joining the points A(2, 1) and B(8, 5) divide the point P(5, 3)?

If A(0, 0) is at the origin and B(10, 0), what point divides AB in the ratio 1:4?

What is the distance formula's application when finding the midpoint between A(5, 5) and B(11, 9)?

Which of the following defines how a point divides a line segment in the section formula?

If a line segment AB is divided by point P in the ratio 1:2, and the coordinates of A are (1, 2) and B are (4, 6), what are the coordinates of P?

What point divides the segment A(0, 0) and B(6, 8) in the ratio 3:1?

What is the distance between the points A(1, 2) and B(4, 6)?

If point P(3, 4) is at a distance of 5 units from the origin, what is P's location?

What is the distance between points A(2, 2) and B(5, 5)?

If C(-1, -2) and D(3, 4) are two points, what is the distance between them?

Which is true for points (2, 3), (2, 7), and (2, -1)?

What is the distance between the points A(1, 1) and B(1, 5)?

Find the distance between points (-2, -3) and (2, 4).

Which of the following points is equidistant from A(1, 1) and B(3, 3)?

What distance does the point (0, 0) have from (3, 4)?

If P(x, y) is equidistant from A(1, 1) and B(3, 3), what equation must it satisfy?

Find the distance between points P(6, -2) and Q(-4, 10).

What is the coordinate form of a point on the x-axis?

Which set of points forms a right-angled triangle: P(-1, -2), Q(3, 2), R(0, 0)?

How do you find the distance between the points A(2, 3) and B(5, 7)?

Which of the following distances is equal to √50?

Which of the following equations represents a straight line in coordinate geometry?

Determine the condition for a point (x, y) to be on the perpendicular bisector of the segment joining A(2, 3) and B(6, 7).

The coordinates of a point P that divides the segment joining A(3, 4) and B(5, 8) in the ratio 1:2 are what?

What is the relationship between the points (-2, 2), (3, -1), and (1, -1)?

Points A(1, 2), B(4, 6), and C(7, 10) are collinear. What is the sum of any two segments distances greater than the third?

Which formula gives the distance of a point P(x, y) from the origin O(0, 0)?

If points A(1, 2), B(3, 6), and C(5, 4) form a triangle, what type of triangle is it?

Convince me: Is point D(3, 5) equidistant from points A(1, 2) and B(5, 8)?

Which of the following points represents the midpoint of A(2, 3) and B(4, 7)?

Finding the slope of line passing through points A(2, 3) and B(6, 7), what is it?

For which of the following conditions do points P(2, 3), Q(0, 0), and R(-2, -3) form a straight line?

If two points A(2, 1) and B(k, 4) lie on the same vertical line, what should k be?

Determine the relation between x and y if point P(x, y) remains equidistant from points A(3, 7) and B(1, 1).

Find the coordinates of point C dividing the segment AB from A(2, 4) to B(6, 8) in the ratio 2:3.

The mid-point of the line segment joining the points (5, -4) and (6, 4) lies on:

For a point (3, -5), the value of (abscissa - ordinate) is:

The mid-point of a line segment divides the line segment in the ratio:

The distance of the point A(4a, 3a) from x-axis is:

If the distance between the points (4, p) and (1, 0) is 5, then p is equal to:

The distance of the point A(4a, 3a) from x-axis is:

In the given figure, ΔABC is an equilateral triangle. AD is a median of the triangle joining the points A(0, 5√3/2), D(0, 0). Points B and C are in same order:

In the given figure, △ABC is an equilateral triangle. AD is a median of the triangle joining the points A(0, 5√3/2), D(0, 0). Points B and C are, in same order:

The mid-point of the line segment joining the points (5, −4) and (6, 4) lies on:

The mid-point of the line segment joining the points (5, –4) and (6, 4) lies:

The mid-point of a line segment divides the line segment in the ratio:

The distance of a point P(3, –7) from y-axis is:

In a circular park, gates are placed at A(10, 20) and B(50, 50). Find the radius of the circular park.

For a point X(a, b), where b > a > 0, the value of [distance from x-axis - distance from y-axis] is:

The mid-point of a line segment divides the line segment in the ratio:

The distance of the point (2, 3) from the origin is:

The mid-point of the line segment joining points (1, 3) and (1, −3) lies:

Find the distance between the points B(40, 10) and C(40, 50).

Anita started from point A and moved to point E along diagonal AC, then from E to B along diagonal DB, and then moved back to A along BA. Find the total distance travelled by Anita.

The mid-point of the line segment joining points (1, 3) and (1, -3) lies:

The distance of the point (2, 3) from the origin is:

The distance of the point (2, 3) from the origin is:

The mid-point of the line segment joining points (1, 3) and (1, −3) lies:

If point (1, 2) is the mid-point of the line segment joining the points (3, 5) and (2a, b), then (a, b) =

The distance of point (–3, 4) from y-axis is:

Find the radius of the semicircular park, where A(2, 3) and B(22, 3) are endpoints of the diameter.

The distance of point (–3, 4) from y-axis is:

If point (1, 2) divides the line segment joining the points (3, 5) and (2p, q) in the ratio 1 : 1, then (p, q) is equal to:

Find the radius of the semicircular park, where A(2, 3) and B(22, 3) are endpoints of the diameter.

If point (a, 2b) is the mid-point of the line segment joining the points (3, 5) and (−1, −1), then (a, b) is equal to:

The distance of point (−3, 4) from y-axis is:

The distance between the points (-4, 2) and (1, 0) is:

The distance between the points (-5, 1) and (2, 2) is:

The distance between the points (–2, 5) and (5, –2) is

The distance between the points (–2, 5) and (5, –2) is

The distance between the points (-2, 5) and (5, -2) is

The fourth vertex D of a parallelogram ABCD whose three vertices are A(–4, 1), B(4, 5) and C(6, 1) is:

The ratio in which the x-axis divides the line segment joining the points A(–8, 4) and B(–6, –2) is:

The line segment joining the points P(-4, -2) and Q(10, 4) is divided by y-axis in the ratio

The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ) is

In the given figure, a circle is centred at (1, 2). The diameter of the circle is

A circle centred at (-1, 2) passes through the point (0, 3). Radius of the circle is

In the given figure, a circle is centred at (1, 2). The diameter of the circle is

The mid-point of the line segment joining the points P(−4, 5) and Q(4, 6) lies on:

Single Answer MCQ

Q-00208117

View explanation

Coordinate Geometry Practice Worksheets

Download and practice Coordinate Geometry worksheets to improve problem-solving accuracy and speed for CBSE Class 10 Mathematics exams.

Coordinate Geometry - Practice Worksheet

This worksheet covers essential long-answer questions to help you build confidence in Coordinate Geometry from Mathematic for Class 10 (Mathematics).

Practice

Questions

Explain the concept of the distance formula. How is it derived, and how can it be applied to find the distance between two points in a coordinate plane?

The distance formula is derived from the Pythagorean theorem. For two points P(x₁, y₁) and Q(x₂, y₂), the distance d between them is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²]. For example, if P(1, 2) and Q(4, 6) are two points, then d = √[(4 - 1)² + (6 - 2)²] = √[9 + 16] = √25 = 5. This formula is applicable in various contexts, such as navigation and physics.

What is the midpoint formula? Explain how to derive it and give an example of its application.

The midpoint formula provides the coordinates of the midpoint M of a line segment joining points A(x₁, y₁) and B(x₂, y₂). It is given by M = ((x₁ + x₂)/2, (y₁ + y₂)/2). For instance, if A(2, 3) and B(4, 7) are the endpoints, the midpoint M = ((2 + 4)/2, (3 + 7)/2) = (3, 5). This concept is useful for dividing a segment in half and has applications in geometry and computer graphics.

Discuss the Section Formula. How do we use it to find a point dividing a line segment in a specific ratio?

The Section Formula helps determine the coordinates of a point P dividing the segment joining points A(x₁, y₁) and B(x₂, y₂) in the ratio m:n. The coordinates of P are given by P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). For example, if A(2, 3) and B(4, 5) and we need to find the point dividing AB in the ratio 2:3, P = ((2*4 + 3*2)/(2 + 3), (2*5 + 3*3)/(2 + 3)) = (2.8, 3.6). This formula is particularly useful in dividing segments in geometry.

How can you determine if three given points form a triangle? Provide a method and example for better understanding.

To determine if three points P(x₁, y₁), Q(x₂, y₂), R(x₃, y₃) form a triangle, use the distance formula to find the lengths of the sides PQ, QR, and PR and check if they satisfy the triangle inequality (the sum of any two sides must be greater than the third). For points P(0, 0), Q(3, 4), R(6, 0), we calculate PQ = 5, QR = 5, and PR = 6. Here, 5 + 5 > 6, 5 + 6 > 5, and 6 + 5 > 5 confirm a triangle is formed.

Explain the concept of collinearity of points. Provide a criterion with an example.

Collinearity refers to points lying on a single straight line. A criterion to check for collinearity is through the area of the triangle formed by the points. If the area is zero, the points are collinear. The area can be calculated using the formula Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. For points A(1, 1), B(2, 2), C(3, 3), substituting gives an area of zero, confirming collinearity.

What is the significance of the coordinate plane in geometry? Discuss its application in solving geometric problems.

The coordinate plane provides a systematic way to describe geometric figures using pairs of numerical coordinates, enhancing precision. It allows for the visualization and calculation of distances, midpoints, and slopes. For instance, solving for the intersection of lines uses coordinates to potentially reveal maximum or minimum points in applications such as physics and engineering. The ability to graphically represent functions and relationships significantly aids problem-solving.

Describe how to find the equation of a line given two points. Illustrate with an example.

To find the equation of a line passing through points A(x₁, y₁) and B(x₂, y₂), first compute the slope m = (y₂ - y₁)/(x₂ - x₁). Use point-slope form: y - y₁ = m(x - x₁). For A(2, 3) and B(4, 7), m = (7 - 3)/(4 - 2) = 2. Thus, the equation becomes y - 3 = 2(x - 2) or y = 2x - 1, describing the line passing through the two points.

What are the different types of slopes, and how do they relate to the orientation of lines on a graph?

Slopes indicate a line's steepness and direction. A positive slope indicates the line rises as it moves right, a negative slope indicates it falls, a zero slope means a horizontal line, and an undefined slope represents a vertical line. For instance, a line with points (1, 2) and (3, 6) has a slope of 2 (positive), implying it rises steeply. Understanding slopes is crucial for graphing lines accurately and for identifying parallel and perpendicular lines.

Discuss the significance of the origin in the coordinate plane. How does it affect the coordinates of other points?

The origin, represented as (0, 0) in a coordinate plane, serves as the reference point for all other coordinates. The coordinates of any point are defined in relation to the origin. For example, a point (x, y) in the first quadrant has both coordinates positive, whereas in the second quadrant x is negative. This relative positioning is vital in graphing and understanding transformations in geometry and algebra, such as translations and reflections.

Coordinate Geometry - Mastery Worksheet

This worksheet challenges you with deeper, multi-concept long-answer questions from Coordinate Geometry to prepare for higher-weightage questions in Class 10.

Mastery

Questions

Using the distance formula, calculate the perimeter of the triangle formed by the points A(1, 2), B(4, 6), and C(7, 2). What type of triangle is formed?

First, find the distance AB = √[(4-1)² + (6-2)²] = √(9 + 16) = 5. \nNext, find BC = √[(7-4)² + (2-6)²] = √(9 + 16) = 5. \nLastly, find AC = √[(7-1)² + (2-2)²] = √(36) = 6. \nSum = 5 + 5 + 6 = 16. \nType: Isosceles triangle.

Prove that the points (2, 3), (4, 5), and (6, 3) are collinear. Use the concept of the slope between each pair of points.

Calculate the slope of AB = (5-3)/(4-2) = 1. \nCalculate the slope of BC = (3-5)/(6-4) = -1. \nSince the slopes differ, the points are not collinear.

Find the coordinates of the point that divides the line segment joining A(-1, 2) and B(3, 8) in the ratio 3:1.

Using the section formula: P(x, y) = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)) = ((3*3 + 1*(-1))/(3+1), (3*8 + 1*2)/(3+1)) = (2, 6). \nThus, the coordinates are (2, 6).

Determine the equation of the line that passes through the points (2, 3) and (4, 7). What is its y-intercept?

Slope (m) = (7-3)/(4-2) = 2. \nEquation: y - 3 = 2(x - 2). \nSimplified: y = 2x - 1. \nThe y-intercept = -1.

Is the point (2, -1) closer to the line y = 3x + 4 than the point (0, -3)? Show your reasoning with calculations.

Calculate distance from a point to a line: \nFor (2, -1), d = |3*2 + 1 + 4|/√(3² + 1²) = 7/√10. \nFor (0, -3), d = |4 - 3|/√10 = 1/√10. \nConclusion: point (2, -1) is farther.

Find the coordinates of the point (x, y) equidistant from A(-2, 1) and B(4, -5). Derive an equation.

Set distances equal: √[(x + 2)² + (y - 1)²] = √[(x - 4)² + (y + 5)²]. \nSquaring: (x + 2)² + (y - 1)² = (x - 4)² + (y + 5)². \nSimplifying gives: 6x - 12y + 40 = 0.

Show that the points A(1, 3), B(2, 7), C(3, 11) form an arithmetic progression in terms of their y-coordinates.

Check y-values: 3, 7, 11. AP: 7-3 = 4 and 11 - 7 = 4, hence AP confirms. \nConclusion: validates AP.

Calculate the area of the triangle formed by the vertices (2, 3), (5, 7), and (4, 1).

Using formula: Area = 1/2 |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. \nSubstituting gives 10. \nArea = 10 square units.

Given the points A(1, -1), B(4, -1), and C(4, 3), determine if they can form a right triangle.

Calculate lengths: AB=3, AC=4, BC=4. Validate Pythagorean theorem: 3² + 4² ≠ 4². \nConclusion: Not a right triangle.

Coordinate Geometry - Challenge Worksheet

The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Coordinate Geometry in Class 10.

Challenge

Questions

Analyze the importance of the distance formula in real-world applications, such as navigation or urban planning. How can it impact the design of efficient transportation routes?

Discuss various applications of the distance formula, considering factors like safety and efficiency. Provide examples from real-life scenarios.

Evaluate the relationship between coordinates and the slope of a line. Explain how this relationship can influence the design of linear structures in architecture.

Analyze how slope impacts construction and stability, using supporting examples. Discuss counterarguments based on specific structures.

Discuss how the section formula can be utilized in dividing resources, such as land or infrastructure projects, between two points. What ethical implications might arise from this?

Evaluate the fairness of division, using quantitative examples. Consider perspectives from both parties involved.

Propose a method to determine whether three given points are collinear. How would you apply this in fields such as robotics or computer graphics?

Explain your method with practical examples, while considering the computational efficiency needed in these fields.

Explore how the concept of equidistant points applies to conflict resolution, such as in urban planning for parks or community spaces. What mathematical models can support this?

Examine various models that can ensure fairness and accessibility, discussing their advantages and limitations.

Evaluate the process of proving whether a quadrilateral formed by four given vertices is a rectangle. Discuss its importance in fields like architecture or robotics.

Describe the geometric properties that must be satisfied. Analyze errors that could arise if these properties are overlooked.

Investigate the implications of coordinate transformations in data visualization. How can this understanding enhance analytical reasoning in fields like data science?

Discuss the significance of transformations for clearer insights and decision-making processes, including potential misinterpretations.

Propose a situation where finding the midpoint between two points has practical significance. How does it relate to fairness in resource distribution?

Examine a case where midpoint finding plays a critical role, justifying your perspective with examples.

Analyze a scenario where calculating the distance between points significantly alters a decision-making process. What are the broader implications of this on community welfare?

Discuss the potential outcomes of relying on inaccurate distances, supported by relevant examples.

Evaluate how understanding the distance and section formulas can contribute to advancements in technology, such as augmented reality. What future developments do you foresee?

Speculate on future applications based on current understanding, while acknowledging limitations and challenges.

Coordinate Geometry Formula Sheet

Use this Class 10 Mathematics Coordinate Geometry Formula Sheet for quick revision before school exams and CBSE exams. It brings together the important formulas, key concepts, and worked examples in one place so students can revise faster and download a printable PDF for offline study.

Important Formulas

Distance Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)

d is the distance between points (x₁, y₁) and (x₂, y₂). This formula is derived from the Pythagorean theorem to calculate the distance in a Cartesian plane.

Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

M is the midpoint between points (x₁, y₁) and (x₂, y₂). It finds the center point of a line segment, useful in geometry and various applications.

Section Formula: P(x, y) = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))

P is the point dividing the line joining (x₁, y₁) and (x₂, y₂) in the ratio m:n. This is useful for partitioning segments in given ratios.

Slope of a Line: m = (y₂ - y₁) / (x₂ - x₁)

m is the slope indicating the steepness of the line. It's crucial in defining the angle of a line and analyzing linear relationships.

Equation of a Line (Slope-Intercept Form): y = mx + c

m is the slope, and c is the y-intercept. This form is key to quickly graphing linear equations.

General Form of a Line: Ax + By + C = 0

A, B, and C are constants. This form is a versatile way to represent linear equations and examine their geometric properties.

Distance of a Point from a Line: d = |Ax₀ + By₀ + C| / √(A² + B²)

This formula calculates the perpendicular distance d from the point (x₀, y₀) to the line Ax + By + C = 0, useful in optimization problems.

Area of Triangle: Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

This formula calculates the area based on vertex coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃). It's useful in both geometry and real-world applications.

Area of Quadrilateral: Area = 1/2 |x₁(y₂ - y₄) + x₂(y₃ - y₁) + x₃(y₁ - y₄) + x₄(y₁ - y₂)|

This finds the area by utilizing the vertices coordinates of the quadrilateral. Important for solving geometric problems.

Equidistant Point: |PA| = |PB| yields (x - x₁)² + (y - y₁)² = (x - x₂)² + (y - y₂)²

This defines the locus of points equidistant from two given points A and B. It's key to understanding geometric relationships.

Worked Examples

Linear equation: ax + by + c = 0

This equation represents a straight line and defines relationships in coordinate geometry.

Slope formula from two points: m = (y₂ - y₁) / (x₂ - x₁)

Calculating the slope m provides insights into the angle of inclination of the line segment between points (x₁, y₁) and (x₂, y₂).

Point of intersection (two lines): Solve {y = mx + c₁; y = nx + c₂}

Finding the intersection helps in solving real-world problems where lines converge or meet.

Perpendicular lines: m₁ * m₂ = -1

If two lines are perpendicular, the product of their slopes is -1, indicating the lines meet at right angles.

Distance from origin to point (x, y): d = √(x² + y²)

This formula calculates the straight-line distance from the origin to a point in a Cartesian plane.

Equation of a vertical line: x = k

Describes a line parallel to the y-axis at x = k; useful for identifying specific vertical positions on the coordinate plane.

Equation of a horizontal line: y = k

Describes a line parallel to the x-axis at y = k; useful in defining horizontal boundaries on graphs.

Circle equation: (x - h)² + (y - k)² = r²

Defines a circle with center (h, k) and radius r, foundational for many geometry applications.

Equation of a parabola: y = ax² + bx + c

This represents a quadratic function and demonstrates the application of coordinate geometry in graphing parabolas.

Equation of a hyperbola: (x²/a²) - (y²/b²) = 1

Provides the standard form of a hyperbola, which is another essential conic section in coordinate geometry.

Explore More Coordinate Geometry Resources

Explore more chapter resources to strengthen your understanding and prepare for exams.

Practice Questions Worksheets Revision Notes Flashcards Download PDFs

Coordinate Geometry Frequently Asked Questions

Understand Coordinate Geometry for Class 10 students, covering distance and section formulas and their applications in mathematics and real-world scenarios.

QWhat are the primary axes in Coordinate Geometry?

In Coordinate Geometry, the primary axes are the x-axis and y-axis. The x-axis is horizontal and measures the abscissa (distance from the y-axis), while the y-axis is vertical and measures the ordinate (distance from the x-axis). Each point on the plane can be defined by its coordinates (x, y), where 'x' represents the location on the x-axis and 'y' on the y-axis.

QHow do you find the distance between two points?

To find the distance between two points with coordinates P(x₁, y₁) and Q(x₂, y₂), you can use the Distance Formula: PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]. This formula derives from the Pythagorean theorem, allowing the calculation of the straight-line distance between the two points in a Cartesian plane.

QCan you explain the concept of the Section Formula?

The Section Formula calculates the coordinates of a point that divides a line segment joining two points A(x₁, y₁) and B(x₂, y₂) in a specific ratio m:n. The coordinates of the dividing point P are given by P = (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n). This formula is essential in various applications, including geometry and design.

QWhat does it mean for points to be collinear?

Collinear points are points that lie on the same straight line. To determine collinearity using coordinates, you can use the Distance Formula. If the sum of the distances between any two points is equal to the distance between them, the points are collinear. For example, three points P, Q, and R are collinear if PQ + QR = PR.

QWhat is an abscissa?

The abscissa is the x-coordinate in a pair of coordinates used to define a point on the Cartesian plane. It represents the horizontal distance of the point from the y-axis. For instance, in the point (x, y), 'x' is the abscissa, indicating how far the point is along the x-axis.

QWhat is an ordinate?

The ordinate is the y-coordinate in a pair used to specify a point in the Cartesian coordinate system. It represents the vertical distance of the point from the x-axis. In a coordinate pair (x, y), 'y' is the ordinate, showing how far the point is up or down the y-axis.

QHow is the Distance Formula derived?

The Distance Formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the distance between the two points) is equal to the sum of the squares of the other two sides. By applying this theorem to a right triangle formed by two points (x₁, y₁) and (x₂, y₂), the formula PQ = √[(x₂ - x₁)² + (y₂ - y₁)²] is obtained.

QWhat are practical applications of Coordinate Geometry?

Coordinate Geometry is applied in various fields, including physics for projectile motion analysis, engineering for designing structures, navigation for flight and maritime routes, and even in art for creating balanced compositions. The ability to represent and analyze spatial relationships mathematically makes it a crucial discipline.

QHow do you plot a point on a graph?

To plot a point on a graph, first identify the coordinates of the point, expressed as (x, y). Move along the x-axis to the value of 'x', and from that position, move vertically to the value of 'y'. Mark the point where these two movements intersect to represent the point graphically.

QWhat is the significance of linear equations in Coordinate Geometry?

Linear equations in two variables, such as ax + by + c = 0, represent straight lines in the Cartesian plane. They are fundamental in Coordinate Geometry as they facilitate the graphical representation of relationships between variables, allowing for the visualization and analysis of data.

QHow do you determine if three points form a triangle?

To determine if three points form a triangle, calculate the distances between each pair of points using the Distance Formula. If the sum of the lengths of any two sides is greater than the length of the third side, the points do indeed form a triangle.

QWhat role does Pythagorean theorem play in Coordinate Geometry?

The Pythagorean theorem is integral to Coordinate Geometry, especially in deriving the Distance Formula. It is used to find distances between points on the Cartesian plane, providing a method for calculating straight-line distances based on the right angles formed by the coordinates.

QWhat is the difference between the X and Y coordinates?

In the Cartesian plane, the X coordinate represents horizontal placement (abscissa) while the Y coordinate represents vertical placement (ordinate). Together, they define a unique point on the plane, indicating its location relative to the Cartesian axes.

QCan the Distance Formula be used with negative coordinates?

Yes, the Distance Formula can be used with negative coordinates. The formula PQ = √[(x₂ - x₁)² + (y₂ - y₁)²] will still compute the distance accurately for any two points, regardless of whether their coordinates are positive or negative, as squaring the differences results in non-negative values.

QWhat are coordinates?

Coordinates are pairs of numbers that define a point's position in a two-dimensional space (the Cartesian plane). They are represented as (x, y), where 'x' is the horizontal position and 'y' is the vertical position relative to the axes.

QHow do you find the midpoint of a line segment?

To find the midpoint of a line segment joining two points A(x₁, y₁) and B(x₂, y₂), use the Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). This gives the coordinates of the point that is exactly halfway between A and B.

QWhat is the graphical representation of a linear equation?

The graphical representation of a linear equation in two variables is a straight line on the Cartesian plane. Each point on the line corresponds to a solution of the equation, displaying the relationship between the x and y variables.

QWhat happens if points lie on different quadrants?

When points lie in different quadrants of the Cartesian plane, the Distance Formula can still be applied without modification. The distances can be calculated considering both positive and negative values for X and Y coordinates, reflecting their placement in the respective quadrants.

QIs coordinate geometry relevant for modern technology?

Yes, Coordinate Geometry is extremely relevant for modern technology. It underpins computer graphics, geographic information systems (GIS), robotics, and even video game development, enabling accurate modeling and representation of the spatial relationships crucial in various technological applications.

QHow does one divide a line segment using the Section Formula?

To divide a line segment between two points A(x₁, y₁) and B(x₂, y₂) in the ratio m:n, apply the Section Formula. The coordinates of the dividing point P are P = (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n). This gives the exact location of the point on the segment relative to the specified ratio.

QWhat is the significance of graphing equations?

Graphing equations visually represents the relationships between variables, allowing for easier interpretation and analysis of their interactions. In mathematics, it helps to identify solutions, behavior trends, and relationship patterns between the quantities involved.

QHow do you calculate the coordinate of a point at a specific distance?

To calculate the coordinates of a point at a specified distance from another point, you can apply the distance from the known point along the x and y axes using unit circle trigonometry or directly use the Distance Formula to find permissible coordinates within the given distance.

Reviewed & Verified by Edzy Expert

Academically Reviewed

Dr. Ananya Sharma

CBSE Mathematics Reviewer

Mathematics educator with 12+ years of experience in CBSE curriculum design, content review, and competitive exam preparation.

M.Sc. MathematicsB.Ed.CTET Qualified

Coordinate Geometry PDF Downloads

Download worksheets, revision guides, formula sheets, and the official textbook PDF for Coordinate Geometry.

Coordinate Geometry Official Textbook PDF

Download the official NCERT/CBSE textbook PDF for Class 10 Mathematics.

Official PDFEnglish EditionNCERT Source

Download PDF

Coordinate Geometry Revision Guide

Use this one-page guide to revise the most important ideas from Coordinate Geometry.

Best for1-page chapter recap

Download PDF

Coordinate Geometry Formula Sheet

Download the Coordinate Geometry formula sheet PDF with important formulas, worked examples, and quick revision support for exam preparation.

Best forImportant formulas for quick revision

Download Formula Sheet PDF

Coordinate Geometry Practice Worksheet

Solve basic and application-based questions from Coordinate Geometry.

Best forCore practice set

Download PDF

Coordinate Geometry Mastery Worksheet

Work through mixed Coordinate Geometry questions to improve accuracy and speed.

Best forMixed difficulty set

Download PDF

Coordinate Geometry Challenge Worksheet

Try harder Coordinate Geometry questions that test deeper understanding.

Best forFor deeper problem solving

Download PDF

Coordinate Geometry Question Bank

Download important questions and exam-style prompts from Coordinate Geometry.

Best forPrintable question set

Download PDF

Coordinate Geometry Flashcards

Revise key terms and definitions from Coordinate Geometry with interactive flashcards. Quick recall practice for CBSE Class 10 Mathematics.

These flash cards cover important concepts from Coordinate Geometry in Mathematics for Class 10 (Mathematics).

1/18

What are coordinates?

1/18

Coordinates are numeric values that determine the position of a point in a plane using an ordered pair (x, y).

How well did you know this?

Not at allPerfectly

2/18

What is the abscissa in a coordinate point (x, y)?

2/18

The abscissa is the x-coordinate, representing the distance of the point from the y-axis.

How well did you know this?

Not at allPerfectly

Active

3/18

What is the distance formula?

Active

3/18

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by: PQ = √((x₂ - x₁)² + (y₂ - y₁)²).

How well did you know this?

Not at allPerfectly

4/18

How to find the distance between two points on the x-axis?

4/18

If points are A(a, 0) and B(b, 0), distance AB = |b - a|.

5/18

How to find the distance between two points on the y-axis?

5/18

If points are C(0, c) and D(0, d), distance CD = |d - c|.

6/18

How to find distance between points in different quadrants?

6/18

Use the Pythagorean theorem to calculate the distance from each point's projections on the axes.

7/18

Calculate the distance between P(4, 6) and Q(6, 8).

7/18

Distance PQ = √((6 - 4)² + (8 - 6)²) = √8 = 2√2.

8/18

What does it mean for points to be collinear?

8/18

Points are collinear if they lie on a straight line, verified by distance relationships.

9/18

How to verify if points (1, 1), (2, 2), and (3, 3) are collinear?

9/18

Calculate distances; if the sum of the distances between them equals the third side, they are collinear.

10/18

What is the midpoint formula?

10/18

The midpoint M between two points P(x₁, y₁) and Q(x₂, y₂) is: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

11/18

What is the section formula?

11/18

It calculates the coordinates of a point dividing a segment in a given ratio m:n as: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

12/18

How to find the distance from a point P(x, y) to the origin?

12/18

The distance OP is given by OP = √(x² + y²).

13/18

What are the coordinates of points on the x-axis?

13/18

Points on the x-axis have coordinates of the form (x, 0).

14/18

What are the coordinates of points on the y-axis?

14/18

Points on the y-axis have coordinates of the form (0, y).

15/18

How to determine the type of triangle formed by points (0, 0), (3, 0), and (0, 4)?

15/18

Calculate distances; if the square of the longest side equals the sum of squares of the others, it's a right triangle.

16/18

What is the standard form of the equation of a line?

16/18

The standard form is Ax + By + C = 0, where A, B cannot both be zero.

17/18

What is the importance of coordinate geometry?

17/18

It connects algebra to geometry, enabling the study of geometric figures using algebraic equations.

18/18

Where is coordinate geometry applied?

18/18

It is applied in physics, engineering, navigation, and art for modeling and analyzing shapes.

View all 18 Coordinate Geometry flashcards

Practice Coordinate Geometry with Interactive Duels

Live Academic Duel

Master Coordinate Geometry via Live Academic Duels

Challenge your classmates or test your individual retention on the core concepts of CBSE Class 10 Mathematics (Mathematics). Compete in speed-recall question rounds matched explicitly to the latest syllabus milestones for Coordinate Geometry.

CBSE-aligned questions

Instant speed-recall rounds

Quick, competitive practice on Coordinate Geometry with zero setup.