Chapter 7: Coordinate Geometry

Important questions — the ones the board asked most (20)

1. Q1Case-based4 marksAsked: 2025

   Case Study - 3 38. A field is in the form of a rectangle. The coordinates of the rectangular field ABCD are A(10, 10), B(40, 10), C(40, 50) and D(x, y). Anil and Anita, two friends decided to have a race. Anita started from point A and moved to point E along the diagonal AC, where E is the point of intersection of both the diagonals of ABCD. From point E, she moved to point B along the other diagonal DB and then moved back to point A along BA. While Anil started from point C and ran to point A via D along the boundary of the field. Based on the above information, answer the following questions : (i) Find the coordinates of point E. 1 (ii) Find the distance between the points B and C. 1 (iii) (a) Find the coordinates of point D and the distance BD. 2 OR (b) Find the total distance travelled by Anita. 2

   

   Medha Ma’am

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   Model answer · Mid-point and distance formula

   1. 1.(i) The coordinates of E are (25, 30). In a rectangle the diagonals bisect each other. So E is the mid-point of the diagonal AC, where A is (10, 10) and C is (40, 50). The mid-point of the line segment joining P(x1, y1) and Q(x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). E = ((10 + 40)/2, (10 + 50)/2) E = (50/2, 60/2) E = (25, 30)1 mark
   2. 2.(ii) The distance between P(x1, y1) and Q(x2, y2) is √((x2 − x1)² + (y2 − y1)²). B is (40, 10) and C is (40, 50). BC = √((40 − 40)² + (50 − 10)²) BC = √(0 + 1600) BC = 40 units1 mark
   3. 3.(iii) (a) E(25, 30) is also the mid-point of the other diagonal DB, where D is (x, y) and B is (40, 10). Coordinates of the mid-point of DB = ((x + 40)/2, (y + 10)/2) So, ((x + 40)/2, (y + 10)/2) = (25, 30)1 mark
   4. 4.Comparing the coordinates: (x + 40)/2 = 25, so x + 40 = 50 and x = 10 (y + 10)/2 = 30, so y + 10 = 60 and y = 50 The coordinates of D are (10, 50). BD = √((10 − 40)² + (50 − 10)²) BD = √(900 + 1600) = √2500 BD = 50 units1 mark

   • Write that E is the mid-point of AC because the diagonals of a rectangle bisect each other.
   • For D, write ((x + 40)/2, (y + 10)/2) = (25, 30) first, then compare to get x = 10 and y = 50.
   • Write the unit with each distance: "BC = 40 units" and "BD = 50 units".

   Marks split as in the board’s published marking scheme

2. Q2Case-based4 marksAsked: 2025

   A field is in the form of a rectangle. The coordinates of the rectangular field ABCD are A(10, 10), B(40, 10), C(40, 50) and D(x, y). Anil and Anita, two friends decided to have a race. Anita started from point A and moved to point E along the diagonal AC, where E is the point of intersection of both the diagonals of ABCD. From point E, she moved to point B along the other diagonal DB and then moved back to point A along BA. While Anil started from point C and ran to point A via D along the boundary of the field. Based on the above information, answer the following questions : (i) Find the coordinates of point E. (ii) Find the distance between the points B and C. (iii) (a) Find the coordinates of point D and the distance BD. OR (b) Find the total distance travelled by Anita.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 215 s

   Model answer · Mid-point and distance formula

   1. 1.(i) The coordinates of E are (25, 30). E is the point where the diagonals of the rectangle ABCD meet. The diagonals of a rectangle bisect each other, so E is the mid-point of the line segment joining A(10, 10) and C(40, 50). The mid-point of the line segment joining P(x1, y1) and Q(x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). E = ((10 + 40)/2, (10 + 50)/2) E = (25, 30)1 mark
   2. 2.(ii) The distance between P(x1, y1) and Q(x2, y2) is √((x2 – x1)² + (y2 – y1)²). Here the points are B(40, 10) and C(40, 50). BC = √((40 – 40)² + (50 – 10)²) BC = √(0 + 1600) = √1600 BC = 40 units1 mark
   3. 3.(iii) (a) E is also the mid-point of the other diagonal BD, which joins B(40, 10) and D(x, y). So the mid-point of BD is E: ((x + 40)/2, (y + 10)/2) = (25, 30)0.5 marks
   4. 4.Comparing the coordinates, we get: (x + 40)/2 = 25, so x + 40 = 50 and x = 10 (y + 10)/2 = 30, so y + 10 = 60 and y = 500.5 marks
   5. 5.Therefore, the coordinates of D are (10, 50).0.5 marks
   6. 6.Using the distance formula for B(40, 10) and D(10, 50): BD = √((10 – 40)² + (50 – 10)²) BD = √((–30)² + (40)²) = √(900 + 1600) = √2500 BD = 50 units0.5 marks

   • Say why E is the mid-point: "The diagonals of a rectangle bisect each other", so E is the mid-point of AC and also of BD.
   • In (iii)(a), set the mid-point of BD equal to E, "((x + 40)/2, (y + 10)/2) = (25, 30)", and solve for x and y one at a time.
   • Put "units" after each distance: BC = 40 units and BD = 50 units.

   Marks split as in the board’s published marking scheme

3. Q3Case-based4 marksAsked: 2025

   There is a semicircular park in Aman's society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre O of the park along the diameter AB as shown in the figure below. Based on the above information, answer the following questions : (i) Find the coordinates of point O. (ii) Find the radius of the semicircular park. (iii) (a) One sapling is kept at point C(12, y). Find the coordinates of C. OR (b) One sapling is kept at point P along AB so that PA = 1/3 PB. Find the coordinates of P.

   

   Medha Ma’am

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   Watch it on the board · 85 s

   Model answer · distance, mid-point, section formula

   1. 1.From the figure, A(2, 3) and B(22, 3). O is the mid-point of the diameter AB, so O = ((2 + 22)/2, (3 + 3)/2) = (12, 3).1 mark
   2. 2.Radius = OA = √[(12 − 2)² + (3 − 3)²] = √100 = 10 units.1 mark
   3. 3.(a) C(12, y) lies on the semicircular boundary, so OC = radius = 10. Then √[(12 − 12)² + (y − 3)²] = 10, so (y − 3)² = 100 and y − 3 = ±10, giving y = 13 or y = −7. OR (b) PA = (1/3) PB means PA : PB = 1 : 3, so P divides AB internally in the ratio 1 : 3 (m₁ = 1, m₂ = 3), and by the section formula P = ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)).1 mark
   4. 4.(a) In the figure C lies on the arc below the diameter AB, so y must be less than 3; hence y = −7 and C = (12, −7). OR (b) P = ((1 × 22 + 3 × 2)/4, (1 × 3 + 3 × 3)/4) = (28/4, 12/4) = (7, 3).1 mark

   Marks split from a structure derived from past papers (checked twice); not CBSE’s official scheme

4. Q4Case-based4 marksAsked: 2025

   Aman's society has a semi-circular park. He wants to plant trees on the boundary (perimeter) of this park. As shown in the figure below, there is a borewell along the diameter AB at the centre O of the park. Based on the above information, answer the following questions: (i) Find the coordinates of point O. (ii) Find the radius of the semi-circular park. (iii) (a) A tree is planted at point C(12, y). Find the coordinates of C. OR (b) A tree is planted at point P along the diameter AB such that PA = 1/3 PB. Find the coordinates of point P.

5. Q5Case-based4 marksAsked: 2025

   Gurveer and Aarushi made a robot that paints the path covered by it while moving on a graph paper. Some coordinates of points are marked on it. The robot starts from the origin (0, 0). It stops at the origin after moving in straight lines on the determined points. Aarushi marked points P(8, 6), Q(12, 2) and S(-6, 6) in this order. The path covered by the robot is shown in the figure. Based on the above, answer the following questions : (i) Find the distance OP. (ii) The equation of line segment QS is 2x + 9y = 42. Find the coordinates of the point of intersection of QS and y-axis. (iii) (a) Point R(4.8, y) divides OP in a certain ratio. Find that ratio and the value of y. OR (iii) (b) Using distance formula, show that PQ/OS = 2/3.

   How the model answer begins

   1. (i) Given: O is the origin (0, 0) and P is (8, 6). The distance of a point P(x, y) from the origin is √(x² + y²). OP = √(8² + 6²) = √(64 + 36) = √100 OP = 10 units

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6. Q6Case-based4 marksAsked: 2025

   Gurveer and Aarushi made a robot that paints the path it covers while moving on a graph paper. Some points are marked with coordinates. The robot starts from the origin (0, 0). It stops at the origin after moving in straight lines through the marked points. Aarushi marked points P(8, 6), Q(12, 2) and S(-6, 6) in that order. The path covered by the robot is shown in the figure. Based on the above, answer the following questions : (i) Find the distance OP. (ii) The equation of line segment QS is 2x + 9y = 42. Find the coordinates of the point of intersection of QS and y-axis. (iii) (a) Point R(4.8, y) divides OP in a certain ratio. Find that ratio and the value of y. OR (iii) (b) Using distance formula, show that PQ/OS = 2/3.

   How the model answer begins

   1. (i) O is the origin (0, 0) and P is (8, 6). The distance of a point P(x, y) from the origin is √(x² + y²). OP = √(8² + 6²) = √(64 + 36) = √100 = 10 units. So the distance OP is 10 units.

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7. Q7Case-based4 marksAsked: 2025

   Gurveer and Arushi built a robot that can paint a path as it moves on a graph paper. Some co-ordinate of points are marked on it. It starts from (0, 0), moves to the points listed in order (in straight lines) and ends at (0, 0). Arushi entered the points P(8, 6), Q(12, 2) and S(-6, 6) in order. The path drawn by robot is shown in the figure. Based on the above, answer the following questions : (i) Determine the distance OP. (ii) QS is represented by equation 2x + 9y = 42. Find the co-ordinates of the point where it intersects y - axis. (iii) (a) Point R(4.8, y) divides the line segment OP in a certain ratio, find the ratio. Hence, find the value of y. OR (iii) (b) Using distance formula, show that PQ/OS = 2/3.

   How the model answer begins

   1. (i) Given: the robot starts at the origin O(0, 0), and P is (8, 6). Formula: the distance of a point P(x, y) from the origin is √(x² + y²). Substitution: OP = √(8² + 6²) Calculation: OP = √(64 + 36) = √100 Answer: OP =…

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8. Q8Case-based4 marksAsked: 2024

   Case Study - 3 38. A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure. Based on the above, answer the following questions : (i) If A is taken as origin, what are the coordinates of the vertices of Delta PQR ? (ii) (a) Find distances PQ and QR. OR (b) Find the coordinates of the point which divides the line segment joining points P and R in the ratio 2 : 1 internally. (iii) Find out if Delta PQR is an isosceles triangle.

   How the model answer begins

   1. If A is taken as origin, then from the figure, the coordinates of the vertices of triangle PQR are P(4, 6), Q(3, 2) and R(6, 5).

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9. Q9Case-based4 marksAsked: 2024

   A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure. Based on the above, answer the following questions : (i) If A is taken as origin, what are the coordinates of the vertices of Delta PQR?

   How the model answer begins

   1. When A is taken as the origin, the coordinates of point P are (4, 6). Taking A as the origin means A is given the coordinates (0, 0), and the position of P is read off along the horizontal and vertical grid lines from…

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10. Q10Case-based4 marksAsked: 2024

    The top of a table is hexagonal in shape. On the basis of the information given above, answer the following questions : (i) Write the coordinates of A and B. (ii) Write the coordinates of the mid-point of line segment joining C and D. (iii) (a) Find the distance between M and Q. OR (iii) (b) Find the coordinates of the point which divides the line segment joining M and N in the ratio 1:3 internally.

    How the model answer begins

    1. (i) Reading the vertices of the hexagonal table top from the grid, the coordinates of A and B are: A(1, 9) and B(5, 13).

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11. Q11Case-based4 marksAsked: 2024

    Case Study - 2 Resident Welfare Association (RWA) of Gulmohar Society in Delhi, have installed three electric poles A, B and C in the society's common park. Despite these three poles, some parts of the park are still in the dark. So, RWA decides to have one more electric pole D in the park. The park can be modelled as a coordinate system given below. Based on the above information, answer the following questions : (i) What is the location of pole C ? (ii) What is the distance of pole B from the corner O of the park ? (iii) (a) Find the location of the fourth pole D such that the four points A, B, C and D form a parallelogram ABCD. OR (b) Find the distance between pole A and pole C.

12. Q12Case-based4 marksAsked: 2024

    Resident Welfare Association (RWA) of Gulmohar Society in Delhi, have installed three electric poles A, B and C in the society's common park. Despite these three poles, some parts of the park are still in the dark. So, RWA decides to have one more electric pole D in the park. The park can be modelled as a coordinate system given below. On the basis of the above information, answer the following questions : (i) What is the position of the pole C ? (ii) What is the distance of the pole B from the corner O of the park ? (iii) (a) Find the position of the fourth pole D so that the four points A, B, C and D form a parallelogram ABCD. OR (b) Find the distance between poles A and C.

    How the model answer begins

    1. (i) The position of the pole C is C(5, 4). This means C is 5 units along the x-axis and 4 units along the y-axis from the corner O(0, 0) of the park.

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13. Q13Case-based4 marksAsked: 2023

    Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O. Based on the above information, answer the following questions : (i) Taking O as origin, coordinates of P are (-200, 0) and of Q are (200, 0). PQRS being a square, what are the coordinates of R and S ? (ii) (a) What is the area of square PQRS ? OR (b) What is the length of diagonal PR in square PQRS ? (iii) If S divides CA in the ratio K:1, what is the value of K, where point A is (200, 800) ?

    How the model answer begins

    1. (i) The coordinates are R(200, 400) and S(–200, 400). Side PQ = 200 – (–200) = 400 units. In square PQRS, R lies 400 units directly above Q(200, 0), and S lies 400 units directly above P(–200, 0).

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14. Q14Case-based4 marksAsked: 2023

    Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle. One such campaign board made by class X student of the school is shown in the figure. Based on the above information, answer the following questions : (i) Find the coordinates of the point of intersection of diagonals AC and BD. (ii) Find the length of the diagonal AC. (iii) (a) Find the area of the campaign Board ABCD. OR (b) Find the ratio of the length of side AB to the length of the diagonal AC.

    How the model answer begins

    1. (i) Given: the vertices of the rectangular campaign board are A(1, 1), B(7, 1), C(7, 5) and D(1, 5). The diagonals of a rectangle bisect each other, so diagonals AC and BD meet at the mid-point of AC. The mid-point of…

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15. Q15Case-based4 marksAsked: 2024

    Case Study - 2 A garden is in the shape of a square. The gardener grew saplings of Ashoka tree on the boundary of the garden at the distance of 1 m from each other. He wants to decorate the garden with rose plants. He chose a triangular region inside the garden to grow rose plants. In the above situation, the gardener took help from the students of class 10. They made a chart for it which looks like the given figure. Based on the above, answer the following questions : (i) If A is taken as origin, what are the coordinates of the vertices of Δ PQR ? 1 (ii) (a) Find distances PQ and QR. 2 OR (b) Find the coordinates of the point which divides the line segment joining points P and R in the ratio 2 : 1 internally. 2 (iii) Find out if Δ PQR is an isosceles triangle. 1

    How the model answer begins

    1. (i) With A as origin: P(4, 6), Q(3, 2), R(6, 5).

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16. Q16Case-based4 marksAsked: 2025

    In a society, there is a circular park having two gates. The gates are placed at points A(10, 20) and B(50, 50), as shown in the figure below. Two fountains are installed at points P and Q on AB such that AP = PQ = QB. Based on the above information, answer the following questions : (i) Find the coordinates of the centre C. 1 (ii) Find the radius of the circular park. 1 (iii) (a) Find the coordinates of the point P. 2 OR (b) Find the distance of the fountain at Q from gate A. 2

    How the model answer begins

    1. (i) The gates A(10, 20) and B(50, 50) are the ends of a diameter of the circular park, so the centre C is the mid-point of the line segment AB. The mid-point of the line segment joining the points P(x1, y1) and Q(x2,…

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17. Q17Case-based4 marksAsked: 2025

    In a society, there is a circular park having two gates. The gates are placed at points A(10, 20) and B(50, 50), as shown in the figure below. Two fountains are installed at points P and Q on AB such that AP = PQ = QB. [figure] On the basis of the above information, answer the following questions : (i) Find the coordinates of centre C. 1 (ii) Find the radius of the circular park. 1 (iii) (a) Find the coordinates of point P. 2 OR (b) Find the distance of the fountain installed at Q from gate A. 2

    How the model answer begins

    1. (i) The gates A(10, 20) and B(50, 50) are the ends of a diameter of the circular park, so the centre C is the mid-point of the line segment AB. Mid-point of the line segment joining (x1, y1) and (x2, y2) = ((x1 + x2)/2,…

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18. Q18Case-based4 marksAsked: 2024

    Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below : Based on the above, answer the following questions : (i) Find the mid-point of the segment joining F and G. (ii) (a) What is the distance between the points A and C ? OR (b) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (iii) What are the coordinates of the point D ?

19. Q19Case-based4 marksAsked: 2024

    Case Study - 3 Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So he started to sketch his own rocket designs on the graph sheet. One such design is given below : Based on the above, answer the following questions : (i) Find the mid-point of the segment joining F and G. (ii) (a) What is the distance between the points A and C ? OR (b) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (iii) What are the coordinates of the point D ?

    How the model answer begins

    1. From the graph, F = (−3, 0) and G = (1, 4). Mid-point of FG = ((−3 + 1)/2, (0 + 4)/2) = (−1, 2).

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20. Q20Case-based4 marksAsked: 2023

    Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle. One such campaign board made by class X student of the school is shown in the figure. Based on the above information, answer the following questions : (i) Find the coordinates of the point of intersection of diagonals AC and BD. (1) (ii) Find the length of the diagonal AC. (1) (iii) (a) Find the area of the campaign Board ABCD. (2) OR (b) Find the ratio of the length of side AB to the length of the diagonal AC. (2)

    How the model answer begins

    1. (i) Given: the campaign board is the rectangle ABCD with A(1, 1), B(7, 1), C(7, 5) and D(1, 5). The diagonals of a rectangle bisect each other. So the point where AC and BD intersect is the mid-point of AC. The…

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83 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

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

   • (A) x-axis ✓
   • (B) y-axis
   • (C) origin
   • (D) neither x-axis nor y-axis

   Answer: (A)

2. 2. CBSE 2026

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

   • (A) 3a ✓
   • (B) - 3a
   • (C) 4a
   • (D) - 4a

   Answer: (A)

3. 3. CBSE 2026

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

   • (A) ± 4 ✓
   • (B) 4
   • (C) - 4
   • (D) 0

   Answer: (A)

4. 4. CBSE 2026

   In the given figure, triangle ABC is an equilateral triangle. AD is the median joining points A(0, 5*sqrt(3)/2) and D(0, 0). The coordinates of points B and C (in that order) are :

   • (A) (- 5, 0), (5, 0)
   • (B) (- 5/2, 0), (5/2, 0) ✓
   • (C) (- 10, 0), (10, 0)
   • (D) (- 5*sqrt(3), 0), (5*sqrt(3), 0)

   Answer: (B)

5. 5. CBSE 2026

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

   • (A) (- 5, 0), (5, 0)
   • (B) (- 5/2, 0), (5/2, 0) ✓
   • (C) (- 10, 0), (10, 0)
   • (D) (- 5 sqrt(3), 0), (5 sqrt(3), 0)

   Answer: (B)

6. 6. CBSE 2026

   In the given figure, triangle ABC is an equilateral triangle. AD is the median joining the points A(0, 5sqrt(3)/2) and D(0, 0). The coordinates of points B and C (in that order) are :

   • (A) (- 5, 0), (5, 0)
   • (B) (- 5/2, 0), (5/2, 0) ✓
   • (C) (- 10, 0), (10, 0)
   • (D) (- 5sqrt(3), 0), (5sqrt(3), 0)

   Answer: (B)

7. 7. CBSE 2026

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

   • (A) sqrt(13) units
   • (B) 3 units
   • (C) 9 units
   • (D) sqrt(29) units ✓

   Answer: (D)

8. 8. CBSE 2026

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

   • (A) sqrt(5) units
   • (B) sqrt(11) units ✓
   • (C) sqrt(13) units
   • (D) sqrt(7) units

   Answer: (B)

9. 9. CBSE 2026

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

   • (A) 2:5 ✓
   • (B) 1:2
   • (C) 2:1
   • (D) 5:2

   Answer: (A)

10. 10. CBSE 2026

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

    • (A) 2 : 5 ✓
    • (B) 1 : 2
    • (C) 2 : 1
    • (D) 5 : 2

    Answer: (A)

Questions parents and students ask

How many marks does "Coordinate Geometry" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 6 marks and about 3 questions a year; by marks it ranks 4 in Maths.

Which are the most important (VVI) questions of "Coordinate Geometry"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 103 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.