Coordinate Geometry — CBSE Class 10 Maths Important Questions
14 CBSE Class 10 Maths practice questions on Coordinate Geometry, each with a full model answer, covering 5 topics from the chapter in the question formats used in the exam.
By The Classmate AI Editorial Team
Reviewed by Classmate AI Team · 30 September 2026
Last updated 25 October 2026
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Coordinate Geometry — CBSE Class 10 Maths Important Questions
Distance, Section, Midpoint — Sorted
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Start your Freemium planThis set covers the rationalised Coordinate Geometry syllabus: the distance formula √((x₂ - x₁)² + (y₂ - y₁)²) and the section formula (including midpoint and trisection), applied to distances, ratios of division and geometric-shape problems.
About Coordinate Geometry
Coordinate Geometry is a formula-driven chapter. Following the rationalised NCERT syllabus, these questions focus on the distance formula and the section formula (midpoint, ratio of division and points of trisection), together with their use in identifying shapes and solving case-based problems.
Key concepts & formulas
The distance between P(x₁, y₁) and Q(x₂, y₂) is PQ = √((x₂ - x₁)² + (y₂ - y₁)²). In particular, the distance of a point (x, y) from the origin is √(x² + y²).
The point dividing the join of (x₁, y₁) and (x₂, y₂) internally in the ratio m₁ : m₂ is ((m₁ x₂ + m₂ x₁)/(m₁ + m₂), (m₁ y₂ + m₂ y₁)/(m₁ + m₂)).
The midpoint (ratio 1 : 1) is ((x₁ + x₂)/2, (y₁ + y₂)/2). The points of trisection are found using the ratios 1 : 2 and 2 : 1.
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Important questions with answers
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Multiple-choice questions (1 mark)
The distance of the point P(3, 4) from the origin is:
- (a)
3 units
- (b)
4 units
- (c)
5 units
- (d)
7 units
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Answer: (c) 5 units.
Distance from the origin = √(x² + y²) = √(3² + 4²) = √(9 + 16) = √25 = 5 units.
The midpoint of the line segment joining A(2, 3) and B(4, 7) is:
- (a)
(3, 5)
- (b)
(2, 5)
- (c)
(3, 4)
- (d)
(6, 10)
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Answer: (a) (3, 5).
Midpoint = ((2 + 4)/2, (3 + 7)/2) = (3, 5).
The point on the x-axis which is equidistant from (2, -5) and (-2, 9) is:
- (a)
(-7, 0)
- (b)
(7, 0)
- (c)
(0, -7)
- (d)
(2, 0)
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Answer: (a) (-7, 0).
Let the point be P(x, 0). Then PA² = PB²: (x - 2)² + 25 = (x + 2)² + 81. Expanding: -4x + 25 = 4x + 81 ⇒ -8x = 56 ⇒ x = -7.
The triangle with vertices A(0, 6), B(-5, 3) and C(3, 1) is:
- (a)
equilateral
- (b)
scalene
- (c)
right-angled and isosceles
- (d)
obtuse-angled
Show model answer
Answer: (c) right-angled and isosceles.
AB² = (-5 - 0)² + (3 - 6)² = 25 + 9 = 34, AC² = (3 - 0)² + (1 - 6)² = 9 + 25 = 34 and BC² = (3 + 5)² + (1 - 3)² = 64 + 4 = 68.
So AB = AC (isosceles), and AB² + AC² = 34 + 34 = 68 = BC², so the angle at A is a right angle.
Assertion–Reason questions (1 mark)
Assertion (A): The point (0, -3) lies on the y-axis.
Reason (R): The x-coordinate of every point lying on the y-axis is zero.
- (a)
Both A and R are true and R is the correct explanation of A
- (b)
Both A and R are true but R is not the correct explanation of A
- (c)
A is true but R is false
- (d)
A is false but R is true
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Answer: (a) Both A and R are true and R is the correct explanation of A.
Every point on the y-axis has x-coordinate 0. Since (0, -3) has x-coordinate 0, it lies on the y-axis, exactly as R explains.
Very short answer questions (2 marks)
Find the value(s) of y for which the distance between the points P(2, -3) and Q(10, y) is 10 units.
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PQ² = (10 - 2)² + (y + 3)² = 100 ⇒ 64 + (y + 3)² = 100 ⇒ (y + 3)² = 36.
⇒ y + 3 = ± 6 ⇒ y = 3 or y = -9.
Find the coordinates of the point which divides the line segment joining (-1, 7) and (4, -3) internally in the ratio 2 : 3.
Show model answer
Using the section formula with m₁ : m₂ = 2 : 3:
x = (2(4) + 3(-1))/(2 + 3) = (8 - 3)/5 = 1, y = (2(-3) + 3(7))/5 = (-6 + 21)/5 = 3.
The point is (1, 3).
Short answer questions (3 marks)
Find the ratio in which the line segment joining A(1, -3) and B(4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
Show model answer
On the x-axis, y = 0. Let the ratio be k : 1. Using the y-coordinate: 0 = (5k - 3)/(k + 1) ⇒ 5k = 3 ⇒ k = 3/5.
So the ratio is 3 : 5.
x = (3(4) + 5(1))/(3 + 5) = (12 + 5)/8 = 17/8.
The point of division is (17/8, 0).
Find the coordinates of the points of trisection of the line segment joining A(4, -1) and B(-2, -3). (NCERT)
Show model answer
The points of trisection P and Q divide AB in the ratios 1 : 2 and 2 : 1.
P (ratio 1 : 2): x = (1(-2) + 2(4))/3 = 6/3 = 2, y = (1(-3) + 2(-1))/3 = -5/3. So P(2, -5/3).
Q (ratio 2 : 1): x = (2(-2) + 1(4))/3 = 0, y = (2(-3) + 1(-1))/3 = -7/3. So Q(0, -7/3).
Show that the points A(1, 7), B(4, 2), C(-1, -1) and D(-4, 4) are the vertices of a square.
Show model answer
AB = √((4 - 1)² + (2 - 7)²) = √(9 + 25) = √34.
BC = √((-1 - 4)² + (-1 - 2)²) = √(25 + 9) = √34.
CD = √((-4 + 1)² + (4 + 1)²) = √(9 + 25) = √34.
DA = √((1 + 4)² + (7 - 4)²) = √(25 + 9) = √34.
All four sides are equal. The diagonals: AC = √((-1 - 1)² + (-1 - 7)²) = √(4 + 64) = √68 and BD = √((-4 - 4)² + (4 - 2)²) = √(64 + 4) = √68.
Since all four sides are equal and the two diagonals are equal, ABCD is a square.
Using the distance formula, show that the points A(1, -1), B(2, 1) and C(4, 5) are collinear.
Show model answer
AB = √((2 - 1)² + (1 + 1)²) = √(1 + 4) = √5.
BC = √((4 - 2)² + (5 - 1)²) = √(4 + 16) = √20 = 2√5.
AC = √((4 - 1)² + (5 + 1)²) = √(9 + 36) = √45 = 3√5.
Since AB + BC = √5 + 2√5 = 3√5 = AC, the point B lies on the segment AC, so A, B and C are collinear.
Long answer questions (5 marks)
If the points A(-2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of the sides AB and BC.
Show model answer
In a parallelogram the diagonals bisect each other, so the midpoint of AC equals the midpoint of BD.
Midpoint of AC = ((-2 + 4)/2, (1 + b)/2) = (1, (1 + b)/2).
Midpoint of BD = ((a + 1)/2, (0 + 2)/2) = ((a + 1)/2, 1).
Equating coordinates: (a + 1)/2 = 1 ⇒ a = 1; and (1 + b)/2 = 1 ⇒ b = 1.
So the vertices are A(-2, 1), B(1, 0), C(4, 1), D(1, 2).
AB = √((1 + 2)² + (0 - 1)²) = √(9 + 1) = √10 units.
BC = √((4 - 1)² + (1 - 0)²) = √(9 + 1) = √10 units.
A(-2, 2) and B(2, 8) are two points.
(i) Find the coordinates of the points that divide AB into four equal parts.
(ii) Show that the middle one of these points is the midpoint of AB.
(iii) Find the length of AB.
Show model answer
(i) The three points P, Q and R divide AB in the ratios 1 : 3, 1 : 1 and 3 : 1.
P (ratio 1 : 3): ((1(2) + 3(-2))/4, (1(8) + 3(2))/4) = (-1, 7/2).
Q (ratio 1 : 1): ((-2 + 2)/2, (2 + 8)/2) = (0, 5).
R (ratio 3 : 1): ((3(2) + 1(-2))/4, (3(8) + 1(2))/4) = (1, 13/2).
(ii) The midpoint of AB is ((-2 + 2)/2, (2 + 8)/2) = (0, 5), which is the point Q.
(iii) AB = √((2 + 2)² + (8 - 2)²) = √(16 + 36) = √52 = 2√13 units.
Case-based questions (4 marks)
Case Study: In a classroom activity, three friends Ayush, Bhavya and Chetan are seated at points A(3, 4), B(6, 7) and C(9, 4) respectively on a coordinate grid marked on the floor (each unit = 1 m). The teacher stands at point T, the midpoint of AC.
(i) Find the distance between Ayush and Bhavya (AB).
(ii) Find the coordinates of the teacher's position T, the midpoint of AC.
(iii) Find the distance BT, OR find the coordinates of the point that divides AB internally in the ratio 1 : 2.
Show model answer
(i) AB = √((6 - 3)² + (7 - 4)²) = √(9 + 9) = √18 = 3√2 m.
(ii) T = ((3 + 9)/2, (4 + 4)/2) = (6, 4).
(iii) BT = √((6 - 6)² + (7 - 4)²) = √(0 + 9) = 3 m.
OR Point dividing AB in the ratio 1 : 2: ((1(6) + 2(3))/3, (1(7) + 2(4))/3) = (12/3, 15/3) = (4, 5).
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Frequently asked questions
What are the important questions on Coordinate Geometry for Class 10?
The distance formula (finding the distance between two points, or checking if three points are collinear) and the section formula (finding a point that divides a line segment in a given ratio, including the midpoint as a special case) are the two most-tested ideas. A smaller but recurring question type asks for the area of a triangle given its three vertices, using the coordinate geometry area formula.What are the important questions in Coordinate Geometry Class 10?
Distance-formula questions (an unknown coordinate, equidistant points, proving a square or collinearity) and section-formula questions (a point in a given ratio, the ratio an axis divides a segment in, trisection and four equal parts). This page has 14 with answers, including a classroom seating case study on Ayush, Bhavya and Chetan.Is the area of a triangle formula still in CBSE Class 10 Coordinate Geometry?
No. The rationalised NCERT chapter covers the distance formula and the section formula (with midpoints and points of trisection) only. Collinearity is therefore shown with distances — AB+BC=AC — as in one of this page's 3-mark questions, not with the area formula.How do I find the ratio in which the x-axis divides a line segment?
Any point on the x-axis has y=0, so let the ratio be k:1 and set the section-formula y-coordinate to zero. For A(1,-3) and B(4,5): (5k-3)/(k+1)=0, so k=3/5, a ratio of 3:5, and the point is (17/8,0).How do you prove four points form a square in coordinate geometry?
Find all four sides and both diagonals with the distance formula. A square needs four equal sides and equal diagonals; equal sides alone give only a rhombus. For A(1,7), B(4,2), C(-1,-1) and D(-4,4) each side is √34 and each diagonal √68, so ABCD is a square, one of this page's 3-mark questions.Where can I find Coordinate Geometry Class 10 notes?
This page's Key concepts & formulas section serves as Coordinate Geometry notes: the distance formula, the section formula, and the midpoint and trisection points as special cases. For the textbook's Exercises 7.1 and 7.2, Classmate AI has a separate NCERT Solutions page for Chapter 7.Where can I get Coordinate Geometry Class 10 extra questions and case studies?
The page goes past NCERT with a parallelogram whose missing coordinates a and b must be found, the points dividing A(-2,2) to B(2,8) into four equal parts, collinearity of (1,-1), (2,1) and (4,5) by distances, and a 4-mark case study on three friends' seats, all with full answers.Where can I practise Coordinate Geometry Class 10 MCQs?
The opening five questions are objective: the distance of (3,4) from the origin, a midpoint, the x-axis point equidistant from (2,-5) and (-2,9), the type of triangle on three vertices, and an assertion–reason item on points of the y-axis. Create Test, free after sign-up, generates timed sets on the chapter.
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