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About Coordinate Geometry — Class 10 ICSE

Find distance, midpoint, section formula, and slope of lines in the coordinate plane. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 3). On this page you can practice 47 questions across three difficulty levels — 19 easy, 18 medium, and 10 hard — each with a visual step-by-step solution, plus a timed 26-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Coordinate Geometry

  • Introduction to Coordinate Geometry: The Cartesian Plane
  • The Distance Formula
  • Midpoint and Section Formulae
  • Slope of a Line
  • Comprehensive Review and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Coordinate Geometry — solved examples for Class 10 ICSE

Example 1easy

Which of the following statements about the distance between two points P(x₁, y₁) and Q(x₂, y₂) in a coordinate plane is TRUE?
  1. A)The distance PQ is always negative if x₁ > x₂.
  2. B)The distance PQ can be calculated as √( (x₂ - x₁)² + (y₁ - y₂)² ).
  3. C)The distance PQ represents the length of the shortest path connecting P and Q.
  4. D)The distance PQ is undefined if x₁ = x₂ and y₁ = y₂.

Step-by-step solution

  1. The distance formula calculates the length of the straight line segment connecting two points. This straight line is always the shortest path between any two points in Euclidean geometry.
  2. Option A is false because distance is always non-negative. Option B is an incorrect variation of the distance formula. Option D is false because if points are identical, the distance is 0, which is defined.

Answer: The distance PQ represents the length of the shortest path connecting P and Q.

Example 2medium

Find the coordinates of a point on the x-axis which is equidistant from the points A(2, -5) and B(-2, 9).
  1. A)(7, 0)
  2. B)(-7, 0)
  3. C)(0, 7)
  4. D)(0, -7)

Step-by-step solution

  1. Let the point on the x-axis be P(x, 0). According to the problem, PA = PB. Squaring both sides, PA² = PB².
  2. Using the distance formula, PA² = (x - 2)² + (0 - (-5))² = (x - 2)² + 5².
  3. And PB² = (x - (-2))² + (0 - 9)² = (x + 2)² + (-9)².
  4. Equating them: (x - 2)² + 25 = (x + 2)² + 81. Expanding: x² - 4x + 4 + 25 = x² + 4x + 4 + 81. Simplifying: -4x + 29 = 4x + 85. This gives -8x = 56, so x = -7.
  5. Thus, the coordinates of the point are (-7, 0).

Answer: (-7, 0)

Example 3hard

The vertices of a triangle are A(p, 2), B(3, -2) and C(7, 6). If the triangle ABC is isosceles with AB = AC, find the value of p.
  1. A)A) 3
  2. B)B) 5
  3. C)C) 7
  4. D)D) 9

Step-by-step solution

  1. Calculate AB² using the distance formula: AB² = (p - 3)² + (2 - (-2))² = (p - 3)² + 4² = (p - 3)² + 16.
  2. Calculate AC² using the distance formula: AC² = (p - 7)² + (2 - 6)² = (p - 7)² + (-4)² = (p - 7)² + 16.
  3. Since AB = AC, then AB² = AC². Equating the expressions: (p - 3)² + 16 = (p - 7)² + 16.
  4. Simplify and solve for p: (p - 3)² = (p - 7)² => p² - 6p + 9 = p² - 14p + 49 => -6p + 9 = -14p + 49 => 8p = 40 => p = 5.

Answer: B) 5

Practice questions on Coordinate Geometry

  1. Q1.easy

    What are the coordinates of the midpoint of the line segment joining the points A(-5, 7) and B(3, -1)?
    1. A)(-1, 3)
    2. B)(-2, 6)
    3. C)(1, -3)
    4. D)(8, -8)
    Show answer

    Answer: (-1, 3)

    Hint: Remember that the midpoint formula averages the x-coordinates and the y-coordinates separately.

  2. Q2.easy

    Ravi calculated the distances between three points P(1, 2), Q(4, 6), and R(7, 10). He found PQ = 5 units, QR = 5 units, and PR = 10 units. Based on these distances, he concluded that points P, Q, and R are collinear. Is his conclusion correct and why?
    1. A)Yes, because PQ + QR = PR (5 + 5 = 10).
    2. B)No, because the distances are not equal.
    3. C)Yes, because the points are equally spaced.
    4. D)No, because collinearity requires a common slope, not just distances.
    Show answer

    Answer: Yes, because PQ + QR = PR (5 + 5 = 10).

    Hint: Recall the condition for three points to be collinear when using the distance formula. It involves the sum of two distances equaling the third.

  3. Q3.easy

    A point P divides the line segment joining A(2, -3) and B(-4, 6) internally in the ratio 1:2. What are the coordinates of point P?
    1. A)(0, 0)
    2. B)(0, -1)
    3. C)(-2, 3)
    4. D)(1, -1)
    Show answer

    Answer: (0, 0)

    Hint: Apply the section formula for internal division. Remember that for a ratio m:n, the formula is ( (nx₁ + mx₂)/(m+n) , (ny₁ + my₂)/(m+n) ).

  4. Q4.medium

    A line segment AB is divided internally by a point P in the ratio 2:3. If the coordinates of A are (-1, 7) and B are (4, -3), find the coordinates of P.
    1. A)(1, 3)
    2. B)(2, 1)
    3. C)(1, 4)
    4. D)(3, 2)
    Show answer

    Answer: (1, 3)

    Hint: Recall the section formula for internal division. The coordinates of P(x, y) are given by [(m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)].

  5. Q5.medium

    If the midpoint of the line segment joining the points P(x, -2) and Q(-3, y) is M(-1, 1), find the values of x and y.
    1. A)x = 1, y = 4
    2. B)x = -1, y = 3
    3. C)x = 2, y = 0
    4. D)x = 3, y = -1
    Show answer

    Answer: x = 1, y = 4

    Hint: The midpoint formula states that the coordinates of the midpoint are the average of the coordinates of the endpoints. Set up two separate equations for the x and y coordinates.

  6. Q6.medium

    What is the slope of a line that is parallel to the line passing through the points A(3, -4) and B(-2, 6)?
    1. A)2
    2. B)-2
    3. C)1/2
    4. D)-1/2
    Show answer

    Answer: -2

    Hint: Parallel lines have the same slope. First, find the slope of the line AB using the formula (y₂ - y₁)/(x₂ - x₁).

  7. Q7.hard

    Show that the points A(a, b+c), B(b, c+a), and C(c, a+b) are collinear. Which of the following conditions must always be true for these points to be collinear? (Assume a, b, c are distinct real numbers for a non-trivial case).
    1. A)A) a+b+c = 0
    2. B)B) a-b+c = 0
    3. C)C) Slope of AB = Slope of BC
    4. D)D) (a-b)/(b-c) = (b-c)/(c-a)
    Show answer

    Answer: C) Slope of AB = Slope of BC

    Hint: Points are collinear if the slope between any two pairs of points is the same. Calculate the slopes of AB and BC.

  8. Q8.hard

    A line segment joining A(3, -4) and B(-2, 5) is divided by the y-axis. In what ratio does the y-axis divide the line segment AB?
    1. A)A) 2:3 internally
    2. B)B) 3:2 internally
    3. C)C) 2:3 externally
    4. D)D) 3:2 externally
    Show answer

    Answer: B) 3:2 internally

    Hint: Any point on the y-axis has its x-coordinate equal to 0. Use the section formula for the x-coordinate to find the ratio.

  9. Q9.hard

    The midpoints of the sides of a triangle are (1, -1), (-4, 2), and (-3, -3). What are the coordinates of its vertices?
    1. A)A) (2, -6), (0, 4), (-8, 0)
    2. B)B) (2, 6), (0, -4), (-8, 0)
    3. C)C) (-2, -6), (0, 4), (8, 0)
    4. D)D) (0, -6), (2, -4), (-8, 0)
    Show answer

    Answer: A) (2, -6), (0, 4), (-8, 0)

    Hint: Let the vertices be (x₁, y₁), (x₂, y₂), (x₃, y₃). Set up a system of equations using the midpoint formula for each side.

These are 9 of the 47 questions available for Coordinate Geometry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.