Coordinate Geometry Important Questions.

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1. Q1Easy

   What is the midpoint of the line segment joining (–1, 7) and (4, –3)?

   • A.(1, 2)
   • B.(1.5, 2)
   • C.(2, 1.5)
   • D.(0, 2)
   Answer: (1.5, 2)

   Explanation: Use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

2. Q2Easy

   What is the distance between the points (4, 5) and (7, 6)?

   • A.√10
   • B.√13
   • C.√17
   • D.√21
   Answer: √13

   Explanation: Use the distance formula: √[(7 - 4)² + (6 - 5)²].

3. Q3Easy

   What is the midpoint of the line segment joining (–2, 2) and (2, 8)?

   • A.(0, 3)
   • B.(0, 5)
   • C.(1, 5)
   • D.(2, 4)
   Answer: (0, 5)

   Explanation: Use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).

4. Q4Easy

   What is the distance between the points (4, 3) and (1, 2)?

   • A.√10
   • B.√13
   • C.√17
   • D.√21
   Answer: √13

   Explanation: Use the distance formula: √[(1 - 4)² + (2 - 3)²].

5. Q5Medium

   If the vertices of a parallelogram taken in order are (1, 2), (4, y), (x, 6), and (3, 5), what is the value of x and y?

   • A.A) x = 2, y = 1
   • B.B) x = 0, y = 3
   • C.C) x = 4, y = 2
   • D.D) x = 3, y = 4
   Answer: B) x = 0, y = 3

   Explanation: In a parallelogram, the diagonally opposite vertices have the same midpoint. So, midpoint of (1, 2) and (x, 6) = midpoint of (4, y) and (3, 5). Solving, we get x = 0 and y = 3.

6. Q6Medium

   If A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are vertices of a parallelogram taken in order, what is the value of p?

   • A.A) 5
   • B.B) 6
   • C.C) 7
   • D.D) 8
   Answer: C) 7

   Explanation: The diagonals of a parallelogram bisect each other. So, midpoint of A(6, 1) and C(9, 4) = midpoint of B(8, 2) and D(p, 3). Solving, we get p = 7.

7. Q7Medium

   What is the type of quadrilateral formed by the points (–1, –2), (1, 0), (–1, 2), and (–3, 0)?

   • A.A) Square
   • B.B) Rhombus
   • C.C) Rectangle
   • D.D) Trapezoid
   Answer: C) Rectangle

   Explanation: Check the distances between consecutive points and verify if opposite sides are equal and adjacent sides are perpendicular. All angles are 90 degrees, confirming it is a rectangle.

8. Q8Medium

   Find the coordinates of point A if AB is the diameter of a circle with center (2, –3) and B is (1, 4).

   • A.A) (1, -8)
   • B.B) (3, -2)
   • C.C) (5, 2)
   • D.D) (0, -5)
   Answer: B) (3, -2)

   Explanation: Since AB is the diameter and the center is the midpoint of AB, midpoint of A(x, y) and B(1, 4) = (2, -3). Solving, we get A as (3, -2).

9. Q9Hard

   If points A(–2, –2) and B(2, –4) are given, what are the coordinates of point P such that AP = (3/7)AB and P lies on AB?

   • A.A) (–1, –1)
   • B.B) (2/7, -4/7)
   • C.C) (0, -3)
   • D.D) (4/7, -6/7)
   Answer: B) (2/7, -4/7)

   Explanation: Since AP = (3/7)AB, P divides AB in the ratio 3:4. Using the section formula, P's coordinates are ((3*2 + 4*(-2))/(3+4), (3*(-4) + 4*(-2))/(3+4)) = (2/7, -4/7).

10. Q10Hard

    What is the area of the rhombus with vertices (3, 0), (4, 5), (–1, 4), and (–2, –1) taken in order?

    • A.A) 20 square units
    • B.B) 25 square units
    • C.C) 30 square units
    • D.D) 15 square units
    Answer: B) 25 square units

    Explanation: The diagonals of the rhombus are calculated using the distance formula. Diagonal 1 (between (3, 0) and (-1, 4)) = √[(3 - (-1))² + (0 - 4)²] = 5 units. Diagonal 2 (between (4, 5) and (-2, -1)) = √[(4 - (-2))² + (5 - (-1))²] = 10 units. Area = (1/2) * (5 * 10) = 25 square units.

11. Q11Hard

    If Q(0, 1) is equidistant from P(5, -3) and R(x, 6), what are the possible values of x?

    • A.A) x = 0 or 10
    • B.B) x = -1 or 11
    • C.C) x = 2 or 8
    • D.D) x = 3 or 7
    Answer: B) x = -1 or 11

    Explanation: Using the distance formula, √[(5 - 0)² + (-3 - 1)²] = √[(x - 0)² + (6 - 1)²]. Solving, we get x = -1 or 11.

12. Q12Hard

    What is the type of quadrilateral formed by the points (4, 5), (7, 6), (4, 3), and (1, 2)?

    • A.A) Square
    • B.B) Rectangle
    • C.C) Rhombus
    • D.D) Trapezoid
    Answer: C) Rhombus

    Explanation: Check the distances between consecutive points and verify if all sides are equal and diagonals are perpendicular. All sides are equal, confirming it is a rhombus.

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