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About Coordinate Geometry — Class 10 Olympiad

Apply distance formula, section formula, and area of triangles; solve locus and collinearity problems. This topic is part of the Olympiad Class 10 mathematics syllabus (chapter: Module 7). On this page you can practice 43 questions across three difficulty levels — 14 easy, 19 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.

Coordinate Geometry — solved examples for Class 10 Olympiad

Example 1easy

Three points A(0, 0), B(3, 4), and C(x, 0) form an isosceles triangle where the side AB is equal in length to AC. What is a possible value of x?
  1. A)A) 3
  2. B)B) 4
  3. C)C) 5
  4. D)D) -3

Step-by-step solution

  1. Calculate the square of the distance AB: AB² = (3 - 0)² + (4 - 0)² = 3² + 4² = 9 + 16 = 25.
  2. Calculate the square of the distance AC: AC² = (x - 0)² + (0 - 0)² = x².
  3. Since AB = AC, we have AB² = AC². So, 25 = x².
  4. This implies x = ±5. Among the given options, x = 5 is a possible value.

Answer: C) 5

Example 2medium

A point P(x, y) is equidistant from three given points A(1, 1), B(3, -1), and C(2, 4). What is the sum of the coordinates of P (i.e., x + y)?
  1. A)5/2
  2. B)11/2
  3. C)7/2
  4. D)9/2

Step-by-step solution

  1. Let P(x, y) be the point. Since PA = PB, we have PA² = PB².
  2. (x-1)² + (y-1)² = (x-3)² + (y+1)² => x²-2x+1+y²-2y+1 = x²-6x+9+y²+2y+1 => 4x - 4y - 8 = 0 => x - y = 2 (Equation 1)
  3. Since PB = PC, we have PB² = PC².
  4. (x-3)² + (y+1)² = (x-2)² + (y-4)² => x²-6x+9+y²+2y+1 = x²-4x+4+y²-8y+16 => -2x + 10y - 10 = 0 => x - 5y = -5 (Equation 2)
  5. Solving Equation 1 and Equation 2: From (1), x = y + 2. Substitute into (2): (y+2) - 5y = -5 => -4y = -7 => y = 7/4. Then x = 7/4 + 2 = 15/4.
  6. So, P(15/4, 7/4). The sum of coordinates x + y = 15/4 + 7/4 = 22/4 = 11/2.

Answer: 11/2

Example 3hard

A point P(x, y) lies on the line y = x and is equidistant from A(2, 3) and B(-4, 1). Find the sum x + y.
  1. A)-1
  2. B)0
  3. C)-1/2
  4. D)1/2

Step-by-step solution

  1. Let the point P be (k, k) as it lies on the line y=x.
  2. Since P is equidistant from A(2, 3) and B(-4, 1), we have PA² = PB². Thus, (k-2)² + (k-3)² = (k+4)² + (k-1)².
  3. Expanding and simplifying gives: 2k² - 10k + 13 = 2k² + 6k + 17. This reduces to -10k + 13 = 6k + 17.
  4. Solving for k: -16k = 4, so k = -1/4. The coordinates of P are (-1/4, -1/4). The sum x+y = -1/4 + (-1/4) = -1/2.

Answer: -1/2

Practice questions on Coordinate Geometry

  1. Q1.easy

    A line segment connects point P(2, -3) and point Q(-4, 6). The origin R(0, 0) divides the segment PQ. In what ratio does R divide PQ?
    1. A)A) 1:1
    2. B)B) 1:2
    3. C)C) 2:1
    4. D)D) 2:3
    Show answer

    Answer: B) 1:2

    Hint: Apply the section formula for a point (0,0) dividing a segment with given endpoints (x1, y1) and (x2, y2) in the ratio m:n.

  2. Q2.easy

    If the points A(1, 4), B(3, -2), and C(k, -8) are collinear, what is the value of k?
    1. A)A) -5
    2. B)B) 0
    3. C)C) 3
    4. D)D) 5
    Show answer

    Answer: D) 5

    Hint: For three points to be collinear, the area of the triangle formed by these points must be zero.

  3. Q3.easy

    If the points A(1, 2), B(4, y), C(x, 6), and D(3, 5) are the vertices of a parallelogram taken in order, find the values of x and y.
    1. A)A) x=5, y=4
    2. B)B) x=6, y=3
    3. C)C) x=7, y=2
    4. D)D) x=4, y=3
    Show answer

    Answer: B) x=6, y=3

    Hint: In a parallelogram, the diagonals bisect each other. This means their midpoints coincide.

  4. Q4.medium

    A point P lies on the x-axis. Find the coordinates of P such that the sum of its distances from A(1, 4) and B(5, 2) is minimized.
    1. A)(11/3, 0)
    2. B)(13/4, 0)
    3. C)(3, 0)
    4. D)(7/2, 0)
    Show answer

    Answer: (11/3, 0)

    Hint: Use the reflection principle: Reflect one of the points across the x-axis. The shortest path will be a straight line connecting the other point to the reflected point.

  5. Q5.medium

    The line segment joining points A(2, 3) and B(5, -6) is divided by a point P in the ratio k:1. If P lies on the line x - 2y + 5 = 0, what is the value of k?
    1. A)-1/22
    2. B)1/22
    3. C)22
    4. D)-22
    Show answer

    Answer: -1/22

    Hint: Use the section formula to find the coordinates of P in terms of k. Then substitute these coordinates into the equation of the line to solve for k.

  6. Q6.medium

    The midpoints of the sides AB, BC, and CA of a triangle ABC are D(1, 2), E(3, 4), and F(5, 6) respectively. Find the coordinates of vertex A.
    1. A)(3, 4)
    2. B)(3, 2)
    3. C)(1, 2)
    4. D)(2, 3)
    Show answer

    Answer: (3, 4)

    Hint: Let the vertices be A(x1, y1), B(x2, y2), C(x3, y3). Use the midpoint formula for each side to set up a system of equations.

  7. Q7.hard

    A point P on the x-axis is chosen such that the sum of its distances from A(1, 3) and B(5, 1) is minimum. What are the coordinates of P?
    1. A)(3,0)
    2. B)(4,0)
    3. C)(2,0)
    4. D)(5,0)
    Show answer

    Answer: (4,0)

    Hint: Consider reflecting one of the points across the x-axis. The shortest path between two points is a straight line.

  8. Q8.hard

    In what ratio does the line x + 2y - 5 = 0 divide the line segment joining A(1, -2) and B(4, 7)?
    1. A)8:13
    2. B)13:8
    3. C)2:3
    4. D)3:2
    Show answer

    Answer: 8:13

    Hint: Assume the line divides the segment in the ratio k:1. Use the section formula to find the coordinates of the division point, then substitute them into the line's equation.

  9. Q9.hard

    The vertices of a triangle ABC are A(p, 1), B(1, q), and C(r, -2). If the centroid of the triangle is at the origin (0, 0) and D, E, F are the midpoints of the sides BC, CA, AB respectively, what is the sum of the coordinates of the centroid of triangle DEF?
    1. A)1
    2. B)-1
    3. C)2
    4. D)0
    Show answer

    Answer: 0

    Hint: Recall the fundamental property relating the centroid of a triangle to the centroid of the triangle formed by its midpoints.

These are 9 of the 43 questions available for Coordinate Geometry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.