Coordinate Geometry Class 10
Download the PDF containing Coordinate Geometry Class 10 notes and a mind map for chapter revision. Understand the distance formula, section formula and midpoint formula through clear explanations, solved examples and useful FAQs.
Coordinates turn a diagram into information you can calculate with. A point’s position helps you find distances, locate a midpoint or divide a line segment in a given ratio. Class 10 Coordinate Geometry connects algebra with geometry, making it useful for map-based questions and problems involving triangles and quadrilaterals.
These Coordinate Geometry Class 10 notes support your 2026 revision with a formula sheet, worked examples and practice questions. Read the explanations first, use the Coordinate Geometry Class 10 mind map for recall, and then apply the formulas independently. A labelled sketch helps you check whether an answer fits the situation.
Coordinate Geometry Basics
The Cartesian plane contains two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They meet at the origin O(0, 0). A point P(x, y) is an ordered pair, with the x-coordinate written first.
- Abscissa: The x-coordinate.
- Ordinate: The y-coordinate.
- Point on the x-axis: Its y-coordinate is 0.
- Point on the y-axis: Its x-coordinate is 0.
- Origin: Both coordinates are 0.
| Quadrant | Coordinate Signs | Example |
|---|---|---|
| I | (+, +) | (3, 4) |
| II | (−, +) | (−3, 4) |
| III | (−, −) | (−3, −4) |
| IV | (+, −) | (3, −4) |
Points on the axes do not belong to any quadrant. Also, the distance of P(x, y) from the x-axis is |y|, while its distance from the y-axis is |x|. Distances are non-negative even when coordinates are negative.
Coordinate Geometry Class 10 Formula Sheet
Let A(x1, y1) and B(x2, y2) be two points. Use the following formulas according to the quantity required.
| Quantity | Formula | Use |
|---|---|---|
| Distance AB | √[(x2 − x1)2 + (y2 − y1)2] | Length between two points. |
| Distance from the origin | √(x2 + y2) | Distance between O(0, 0) and P(x, y). |
| Midpoint | ((x1 + x2)/2, (y1 + y2)/2) | Internal division in the ratio 1:1. |
| Internal section formula | ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) | For AP:PB = m:n, with m and n positive. |
Distance Formula Class 10
The distance formula comes from Pythagoras’ theorem. The horizontal and vertical coordinate differences form the perpendicular sides of a right triangle; the segment joining the points is its hypotenuse.
Example 1: Find the Distance Between Two Points
Question: Find the distance between A(−2, 1) and B(4, 9).
AB = √[(4 − (−2))2 + (9 − 1)2]
AB = √(62 + 82)
AB = √100
AB = 10 units
Keep Negative Coordinates in Brackets
Subtracting −2 gives 4 − (−2) = 6, not 2. Write coordinate differences before squaring. Remember that (−3)2 = 9.
Section Formula Class 10
The section formula finds a point that divides a line segment internally in a specified ratio. If AP:PB = m:n, the coefficient m multiplies B’s coordinates and n multiplies A’s coordinates.
Example 2: Divide a Segment in the Ratio 2:1
Question: Find P dividing A(1, 2) and B(7, 8) internally in the ratio AP:PB = 2:1.
x = (2 × 7 + 1 × 1)/(2 + 1)
x = 15/3
x = 5
y = (2 × 8 + 1 × 2)/(2 + 1)
y = 18/3
y = 6
Therefore, P = (5, 6).
P is closer to B because PB is the shorter part of the segment. Its coordinates lie between the corresponding endpoint coordinates.
Midpoint Formula Class 10
A midpoint divides a segment into two equal parts. Substituting m = n = 1 into the section formula gives the average of the x-coordinates and the average of the y-coordinates.
Example 3: Find a Midpoint
Question: Find the midpoint of A(−6, 4) and B(2, −2).
x = (−6 + 2)/2
x = −2
y = (4 + (−2))/2
y = 1
Midpoint = (−2, 1)
Coordinate Geometry Class 10 Questions with Solutions
Example 4: Find a Point on the x-Axis
Question: Find the point on the x-axis equidistant from A(2, 3) and B(6, 5).
Let P = (x, 0).
Since PA = PB, PA2 = PB2.
(x − 2)2 + 9 = (x − 6)2 + 25
x2 − 4x + 13 = x2 − 12x + 61
8x = 48
x = 6
Therefore, P = (6, 0).
Example 5: Find Where a Segment Meets the y-Axis
Question: In what ratio does the y-axis divide the segment joining A(−4, 2) and B(6, 7)? Find the point of intersection.
Let AP:PB = m:n.
On the y-axis, x = 0.
(6m − 4n)/(m + n) = 0
6m = 4n
m/n = 2/3
Therefore, AP:PB = 2:3.
y = (2 × 7 + 3 × 2)/(2 + 3)
y = 20/5
y = 4
Intersection point = (0, 4)
Using Coordinates to Classify a Triangle
Calculate the three squared side lengths to compare sides efficiently. Equal lengths establish an isosceles or equilateral triangle. A Pythagorean equality establishes a right angle when the points form a non-degenerate triangle.
Example 6: Check for a Right-Angled Triangle
Question: Classify the triangle with vertices A(0, 0), B(6, 0) and C(0, 8).
AB2 = 36
AC2 = 64
BC2 = 36 + 64 = 100
AB2 + AC2 = BC2
Therefore, the triangle is right-angled at A.
Coordinate Geometry Class 10 Mind Map and Revision 2026
Use the mind map of Coordinate Geometry Class 10 to connect coordinates, distance, midpoint and internal division. Decide whether a question asks for a length, a location or a ratio before selecting a formula.
For 2026 revision, practise direct calculations first, then move to missing coordinates, axis intersections and geometrical applications. Match your NCERT exercise questions with the textbook edition you use. When consulting older notes containing additional formulas, check the syllabus applicable to your course.
Important Questions for Quick Practice
-
Find the distance of (−5, 12) from the origin.
Answer: 13 units. -
Find the midpoint of (2, −4) and (8, 6).
Answer: (5, 1). -
Find the point dividing (0, 0) and (9, 6) internally in the ratio 1:2.
Answer: (3, 2). -
Find the distance between (−3, 5) and (4, 5).
Answer: 7 units. -
A segment has midpoint (3, 2) and one endpoint (1, −2).
Find the other endpoint.
Answer: (5, 6).
Coordinate Geometry Class 10 MCQ with Answers
1. A point on the y-axis has:
A. x = 0 B. y = 0 C. x = y D. x = 1
Answer: A.
2. The midpoint formula is a special case of:
A. Distance formula B. Section formula
C. Quadratic formula D. Sum formula
Answer: B. The dividing ratio is 1:1.
How to Avoid Common Coordinate Geometry Mistakes
- Keep the order of x- and y-coordinates consistent.
- Write negative values in brackets before substitution.
- State which segment ratio is given, such as AP:PB.
- Remember that distances are non-negative.
- Check both coordinate equations when finding a dividing ratio.
- Use exact surd answers unless an approximation is requested.
Related Notes and Worksheet Practice
Apply these formulas using the Coordinate Geometry Class 10 Worksheet with Solutions . Attempt each question independently before checking the working.
Revise geometrical reasoning with Triangles Class 10 Notes and Mind Map , or explore the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Coordinate Geometry Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Read the explanations first, then use the map to recall the formulas before solving questions.
Which Coordinate Geometry Class 10 formulas should I revise?
Focus on the distance formula, internal section formula and midpoint formula. Also remember coordinate conditions for points on the axes and the distance from the origin.
Why does the section formula multiply the opposite endpoint?
If AP:PB = m:n, then P = A + [m/(m + n)](B − A). Expanding gives (nA + mB)/(m + n), coordinate by coordinate. A larger m places P farther from A and closer to B.
What happens if I reverse the endpoints?
Reverse the ratio as well. If AP:PB = 2:3, then BP:PA = 3:2. Changing the endpoint order without changing the ratio usually produces a different dividing point.
Can distance be negative?
No. Coordinates and coordinate differences may be negative, but distance is non-negative. The distance formula uses the non-negative square root.
Is every point equidistant from two endpoints their midpoint?
No. Many points can be equidistant from two endpoints. The midpoint is the equidistant point that also lies on their segment. Equidistance alone does not establish that a point is the midpoint.
How do I find the points of trisection?
Apply the internal section formula twice, using ratios 1:2 and 2:1 from the same starting endpoint. These points divide the segment into three equal parts.
How can I check whether three points are collinear?
For three distinct points, calculate their pairwise distances. If the greatest distance equals the sum of the other two, the points are collinear. For example, distances 3, 4 and 7 satisfy this condition.
Can a missing-coordinate question have two answers?
Yes. For example, if (x, 0) is 5 units from the origin, then x² = 25 gives x = 5 or −5. Check both values against any additional conditions in the question.
Do I need the area-of-a-triangle formula from older notes?
Check your applicable syllabus before adding older or additional topics to your exam revision. Start with the distance and internal section formulas taught in the current NCERT chapter. A formula’s appearance in an older guide does not establish its current assessment status.
How should I practise Coordinate Geometry previous year questions?
Use questions with a verified board and year. Solve them without looking at the answer, then check formula selection, coordinate signs and interpretation. Include both direct calculations and questions involving unknown values.
Where can I practise Coordinate Geometry questions with solutions?
Visit the Coordinate Geometry Class 10 worksheet with solutions after revising the notes and mind map. Use it to check whether you can apply each formula independently.
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