YOUR STUDY SPACE · CBSE

Coordinate Geometry — Class 10 Mathematics

Mathematics · Chapter 7

Download revision card →

CBSE Class 10 community

View all

All available subjects

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Mathematics
Science
+ 10 more

~7 pages per kitPage 1 free

Get all 12 kits · ₹299

Mathematics (Standard and Basic)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

70+ formulas and theorems
Board-paper chapter counts
24 traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Social Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

History
Economics
+ 2 more

~7 pages per kitPage 1 free

Get all 4 kits · ₹69

English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

First Flight
Footprints without Feet

~7 pages per kitPage 1 free

Get both kits · ₹69

Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Equations and formulas
Book tables as charts
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi A · Kshitij 2 + Kritika 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

15 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Computer Applications 2027

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

HTML & CSS examples
Protocol tables
Cyber ethics

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi B · Sparsh 2 + Sanchayan 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

17 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Sanskrit · Shemushi 2 + Vyakaranavithi + Abhyaswaan Bhav

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

9 पाठ
सन्धि-समास
प्रत्यय

Only 7 pagesPage 1 free

Get the kit · ₹69

Browse all Class 10 kits →

Chapter summary

Coordinate geometry connects algebra with geometry by locating points on a plane using ordered pairs . The chapter introduces the distance formula to find the length between two points, and the section formula to find the coordinates of a point dividing a line segment in a given ratio (including the midpoint as a special case). These tools let students compute lengths, verify shapes and divide segments without drawing, turning geometric questions into straightforward algebraic calculations.

Distance formulaSection formulaMidpoint formula

Key ideas & terms

Distance formula: .
Section formula: Coordinates of a point dividing a segment in ratio .
Midpoint: The point dividing a segment in the ratio .
Coordinate plane: A plane with x- and y-axes for locating points.

Extra questions & answers

Write the distance formula between points and .

Find the midpoint of the segment joining and .

Midpoint .

Write the distance formula between points and .

Distance .

Find the distance between the points and .

units.

Write the section formula for a point dividing a segment in ratio .

The point is .

Write the midpoint formula.

The midpoint of the segment joining and is .

Find the midpoint of the segment joining and .

How can you check whether three points are collinear using coordinates?

They are collinear if the sum of the distances of two pairs equals the third, or if the area of the triangle they form is zero.

In what ratio does the point divide the segment joining and ?

Using the section formula on x-coordinates: gives , so it is the midpoint.

Find the coordinates of the point dividing the join of and in the ratio .

EXERCISE 7.1

1. (i) Find the distance between the following pair of points :

Using the distance formula, Therefore, the distance is units.

1. (ii) Find the distance between the following pair of points :

Using the distance formula, Therefore, the distance is units.

1. (iii) Find the distance between the following pair of points :

Using the distance formula, Therefore, the distance is units.

2. Find the distance between the points and . Can you now find the distance between the two towns A and B discussed in Section 7.2.

The distance is Therefore, the distance between towns A and B is .

3. Determine if the points and are collinear.

Let the points be A, B and C. Their distances are and No one distance equals the sum of the other two. Therefore, the points are not collinear.

4. Check whether and are the vertices of an isosceles triangle.

Let the points be A, B and C. Then and Since , the points are the vertices of an isosceles triangle.

5. In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.

From Fig. 7.8, the coordinates are , , and . The four sides satisfy while the diagonals satisfy Thus, all four sides are equal and the diagonals are equal. Therefore, ABCD is a square, so Champa is correct.

6. (i) Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

Let the points, in order, be A, B, C and D. The four sides are and the diagonals are Since all four sides are equal and both diagonals are equal, the quadrilateral is a square.

6. (ii) Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

The point is the midpoint of the line segment joining and , because Hence, three of the given points are collinear. Therefore, they do not form a quadrilateral.

6. (iii) Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

Let the points, in order, be A, B, C and D. Their side lengths are and Both pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram.

7. Find the point on the x-axis which is equidistant from and .

Let the required point be . Since it is equidistant from the two given points, Thus, which gives . Therefore, the required point is .

8. Find the values of for which the distance between the points and is units.

By the distance formula, Squaring gives so Hence, or . Therefore, or .

9. If is equidistant from and , find the values of . Also find the distances and .

First, Also, Since , giving or . In both cases, units. When , When ,

10. Find a relation between and such that the point is equidistant from the point and .

Equating the squares of the two distances gives On simplifying, Therefore, the required relation is .

EXERCISE 7.2

1. Find the coordinates of the point which divides the join of and in the ratio .

Using the section formula, the required point is Therefore, its coordinates are .

2. Find the coordinates of the points of trisection of the line segment joining and .

The first point divides the segment in the ratio , so its coordinates are The second point divides it in the ratio , so its coordinates are Therefore, the points of trisection are and .

3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of each. flower pots have been placed at a distance of from each other along AD, as shown in Fig. 7.12. Niharika runs th the distance AD on the 2nd line and posts a green flag. Preet runs th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Taking A as the origin, Niharika's flag is at , because . Preet's flag is at , because . The distance between the flags is The midpoint is Therefore, Rashmi should post the blue flag on the fifth line at a distance of , or , from A.

4. Find the ratio in which the line segment joining the points and is divided by .

Let divide the segment in the ratio . Using the x-coordinate, Therefore, so . Hence, The y-coordinate also satisfies this ratio. Therefore, the required ratio is .

5. Find the ratio in which the line segment joining and is divided by the x-axis. Also find the coordinates of the point of division.

Let the x-axis divide AB in the ratio . Since the point of division has y-coordinate , Thus, , so the ratio is . The point is therefore the midpoint of AB:

6. If and are the vertices of a parallelogram taken in order, find and .

The diagonals of a parallelogram bisect each other. Therefore, the midpoint of the first and third vertices equals the midpoint of the second and fourth vertices: Hence, and . Therefore, and .

7. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is and B is .

Let . Since the centre is the midpoint of diameter AB, Thus, and , giving and . Therefore, .

8. If A and B are and , respectively, find the coordinates of P such that and P lies on the line segment AB.

Since , the remaining part is . Thus, . By the section formula,

9. Find the coordinates of the points which divide the line segment joining and into four equal parts.

The three required points lie at , and of the way from A to B. Their coordinates are

10. Find the area of a rhombus if its vertices are and taken in order. [Hint : Area of a rhombus = (product of its diagonals)]

The diagonal joining and has length The other diagonal has length Therefore, the area is Hence, the area of the rhombus is square units.