Coordinate Geometry – Class 10

Coordinate Geometry is one of the most scoring chapters in Class 9, 10 and 11 Mathematics. Students often struggle with concepts like mid-point, section formula, centroid, medians and parallelogram properties. To help you practise better, we have compiled a complete set of high-quality coordinate geometry questions extracted from standard textbooks. These questions cover all important concepts and are perfect for board exam preparation, competitive exams and school tests.

Practice these problems to strengthen your fundamentals and improve accuracy in exams.

1. Find the coordinates of the points which divide the line segment joining the points (–4, 0) and (0, 6) in four equal parts.
2. Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (10, 10).
3. Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).
4. If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.
5. Show that the points A (1, 0), B (5, 3), C (2, 7) and D (–2, 4) are the vertices of a parallelogram.
6. Determine the ratio in which the point P (m, 6) divides the join of A(–4, 3) and B(2, 8). Also, find the value of m.
7. Determine the ratio in which the point (–6, a) divides the join of A(–3, 1) and B(–8, 9). Also find the value of a.
8. ABCD is a rectangle formed by joining the points A (–1, –1), B (–1, 4), C (5, 4) and D (5, –1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square, a rectangle or a rhombus? Justify your answer.
9. Points P, Q and R divide the line segment joining the points A (1, 2) and B (6, 7) in 3 equal parts. Find the coordinates of the points P, Q and R.
10. If A and B are two points having coordinates (–2, –2) and (2, –4) respectively, find the coordinates of P such that AP = 3/7 AB.
11. Find the coordinates of the points which divide the line segment joining A (–2, 2) and B (2, 8) into four equal parts.
12. Three consecutive vertices of a parallelogram are (–2, –1), (1, 0) and (4, 3). Find the fourth vertex.
13. The points (3, –4) and (–6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (–1, –3), find the coordinates of the fourth vertex.
14. If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, –3) and (3, 4), find the vertices of the triangle.
15. Determine the ratio in which the straight line x – y – 2 = 0 divides the line segment joining (3, –1) and (8, 9).
16. Three vertices of a parallelogram are (a + b, a – b), (2a + b, 2a – b), (a – b, a + b). Find the fourth vertex.
17. If two vertices of a parallelogram are (3, 2), (–1, 0) and the diagonals cut at (2, –5), find the other vertices of the parallelogram.
18. If the coordinates of the mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7), find its vertices.
19. The line segment joining the points P (3, 3) and Q (6, –6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
20. If three consecutive vertices of a parallelogram are (1, –2), (3, 6) and (5, 10), find its fourth vertex.
21. If the points A (a, –11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.
22. If the coordinates of the mid-points of the sides of a triangle are (3, –2), (–3, 1) and (4, 3), find the coordinates of its vertices.
23. The line segment joining the points (3, –4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, –2) and (5/3, q) respectively, find the values of p and q.
24. If the line joining the points (2, 1) and (5, –8) is trisected at the points P and Q, find the points P and Q. If point P lies on the line 2x – y + k = 0, find the value of k.
25. A (2, 2), B (6, 5) and C (1, 4) are the vertices of ΔABC.
    (a) The median from A meets BC in D. Find the coordinates of point D.
    (b) Find the coordinates of point P on AD such that AP : PD = 2 : 1.
    (c) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
    (d) What do you observe?
26. If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, find the values of k and p.
27. A point P divides the line segment joining the points A (3, –5) and B (–4, 8) such that AP / PB = k / 8. If P lies on the line x + y = 0, find the value of k.
28. The mid-point P of the line segment joining the points A (–10, 4) and B (–2, 0) lies on the line segment joining the points C (–9, –4) and D (–4, y). Find the ratio in which P divides CD. Also, find the value of y.
29. If the point C (–1, 2) divides internally the line segment joining the points A (2, 5) and B (x, y) in the ratio 3 : 4, find the value of x² + y².
30. ABCD is a parallelogram with vertices A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃). Find the coordinates of the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂ and y₃.
31. The points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the vertices of ΔABC.
    (a) The median from A meets BC at D. Find the coordinates of D.
    (b) Find the coordinates of the point P on AD such that AP : PD = 2 : 1.
    (c) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
    (d) What are the coordinates of the centroid of the triangle ABC?