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About Coordinate Geometry — Class 9 IB

Apply distance and midpoint formulas, find gradients, and write equations of lines. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 3). On this page you can practice 50 questions across three difficulty levels — 10 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Coordinate Geometry — solved examples for Class 9 IB

Example 1easy

Point P(x, y) is located in the second quadrant of the Cartesian plane. Which of the following statements about the coordinates x and y must be true?
  1. A)x > 0 and y > 0
  2. B)x < 0 and y > 0
  3. C)x < 0 and y < 0
  4. D)x > 0 and y < 0

Step-by-step solution

  1. The Cartesian plane is divided into four quadrants. The first quadrant has x > 0, y > 0.
  2. Moving counter-clockwise, the second quadrant has negative x-values and positive y-values.
  3. Therefore, for a point in the second quadrant, x < 0 and y > 0.

Answer: x < 0 and y > 0

Example 2medium

The distance between point A(k, -2) and B(4, 1) is 5 units. What is the positive value of k?
  1. A)A) 0
  2. B)B) 2
  3. C)C) 4
  4. D)D) 8

Step-by-step solution

  1. Apply the distance formula: 5 = √[(4 - k)² + (1 - (-2))²].
  2. Substitute given values and simplify: 5 = √[(4 - k)² + (3)²]. Square both sides: 25 = (4 - k)² + 9.
  3. Solve for k: 16 = (4 - k)². Taking the square root of both sides gives 4 = 4 - k or -4 = 4 - k.
  4. From 4 = 4 - k, we get k = 0. From -4 = 4 - k, we get k = 8. The positive value of k is 8.

Answer: D) 8

Example 3hard

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)(0, -7)
  3. C)(7, 0)
  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, so PA² = PB².
  2. Using the distance formula: (x - 2)² + (0 - (-5))² = (x - (-2))² + (0 - 9)²
  3. This simplifies to (x - 2)² + 5² = (x + 2)² + (-9)², which is x² - 4x + 4 + 25 = x² + 4x + 4 + 81.
  4. Further simplifying: -4x + 29 = 4x + 85. Solving for x gives 8x = -56, so x = -7. The point is (-7, 0).

Answer: (-7, 0)

Practice questions on Coordinate Geometry

  1. Q1.easy

    A student, Ravi, is trying to find the distance between two points A(2, 3) and B(5, 7). His calculation starts as follows: Distance = √((5 - 2)² + (7 - 3)²). Which of the following is the correct next step in his calculation?
    1. A)Distance = √(3² + 4²)
    2. B)Distance = √(7² + 10²)
    3. C)Distance = √(3 + 4)
    4. D)Distance = √(10 - 5)
    Show answer

    Answer: Distance = √(3² + 4²)

    Hint: First, evaluate the differences inside the parentheses before squaring them.

  2. Q2.easy

    The midpoint of a line segment connecting P(x₁, y₁) and Q(x₂, y₂) is given by the formula M = ((x₁ + x₂)/2, (y₁ + y₂)/2). If a student calculated the midpoint of A(1, 5) and B(7, 3) as (4, 8), what was their mistake?
    1. A)They incorrectly added the x-coordinates.
    2. B)They incorrectly subtracted the y-coordinates.
    3. C)They incorrectly added the y-coordinates instead of averaging them.
    4. D)They forgot to divide the x-coordinates by 2.
    Show answer

    Answer: They incorrectly added the y-coordinates instead of averaging them.

    Hint: Carefully apply the midpoint formula to both the x and y coordinates. What operations are involved for each?

  3. Q3.easy

    Consider a line segment that connects the points (2, 5) and (2, -3). Which of the following statements correctly describes the gradient of this line segment?
    1. A)The gradient is positive.
    2. B)The gradient is negative.
    3. C)The gradient is zero.
    4. D)The gradient is undefined.
    Show answer

    Answer: The gradient is undefined.

    Hint: What kind of line is formed when the x-coordinates of two points are the same? Think about the formula for gradient: m = (y₂ - y₁)/(x₂ - x₁).

  4. Q4.medium

    M(1, -2) is the midpoint of the line segment AB. If point A is (-3, 4), what are the coordinates of point B?
    1. A)A) (5, -8)
    2. B)B) (-5, 8)
    3. C)C) (4, 2)
    4. D)D) (-4, -6)
    Show answer

    Answer: A) (5, -8)

    Hint: Let the coordinates of B be (x_B, y_B). Use the midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) for both x and y coordinates.

  5. Q5.medium

    Three points P(1, 5), Q(3, 9), and R(k, 13) are collinear. What is the value of k?
    1. A)A) 4
    2. B)B) 5
    3. C)C) 6
    4. D)D) 7
    Show answer

    Answer: B) 5

    Hint: If three points are collinear, the gradient (slope) between any two pairs of points must be the same.

  6. Q6.medium

    Find the equation of the line that passes through the points (-1, 3) and (2, -3).
    1. A)A) y = -2x + 3
    2. B)B) y = -2x + 1
    3. C)C) y = 2x - 1
    4. D)D) y = -x + 4
    Show answer

    Answer: B) y = -2x + 1

    Hint: First, find the gradient (slope) of the line using the two given points. Then, use the point-slope form (y - y₁ = m(x - x₁)) with either point.

  7. Q7.hard

    If the midpoints of the sides of a triangle are P(1, 2), Q(0, -1), and R(2, -1), what is the sum of the x-coordinates of its vertices?
    1. A)0
    2. B)1
    3. C)2
    4. D)3
    Show answer

    Answer: 3

    Hint: Let the vertices of the triangle be (x1, y1), (x2, y2), and (x3, y3). Use the midpoint formula to set up a system of equations.

  8. Q8.hard

    Consider the statement: The points (a, 0), (0, b) and (1, 1) are collinear if and only if 1/a + 1/b = 1. Which of the following is true?
    1. A)True
    2. B)False
    3. C)Only if a = b
    4. D)Only if a = 1
    Show answer

    Answer: True

    Hint: For three points to be collinear, the slope between any two pairs of points must be equal. Calculate the gradients and set them equivalent.

  9. Q9.hard

    A line L is the perpendicular bisector of the line segment joining the points A(-3, 2) and B(5, 4). Find the equation of line L.
    1. A)x - 4y = 1
    2. B)4x - y = -1
    3. C)4x + y = 7
    4. D)x + 4y = 13
    Show answer

    Answer: 4x + y = 7

    Hint: The perpendicular bisector passes through the midpoint of the segment and has a slope that is the negative reciprocal of the segment's slope.

These are 9 of the 50 questions available for Coordinate Geometry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.