Coordinate Geometry
Class 10 Mathematics - Complete Notes & Practice Questions
Theory & Formulas
Coordinate System
The coordinate system is a method for representing points in a plane using ordered pairs of numbers (x, y).
Key Concepts
- The x-axis and y-axis divide the plane into four quadrants
- The point of intersection (0, 0) is called the origin
- The x-coordinate is called the abscissa
- The y-coordinate is called the ordinate
Distance Formula
The distance between two points (x₁, y₁) and (x₂, y₂) is given by:
Example:
Find the distance between points A(2, 3) and B(5, 7).
Solution: d = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units
Section Formula
Finds the coordinates of a point that divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio m:n.
y = (my₂ + ny₁)/(m + n)
Example:
Find the coordinates of the point which divides the line segment joining A(1, 2) and B(4, 5) in the ratio 2:1.
Solution: x = (2×4 + 1×1)/(2+1) = (8+1)/3 = 3
y = (2×5 + 1×2)/(2+1) = (10+2)/3 = 4
Coordinates: (3, 4)
Midpoint Formula
Finds the coordinates of the midpoint of a line segment joining (x₁, y₁) and (x₂, y₂).
y = (y₁ + y₂)/2
Example:
Find the midpoint of the line segment joining P(3, 4) and Q(5, 8).
Solution: x = (3+5)/2 = 4, y = (4+8)/2 = 6
Midpoint: (4, 6)
Area of a Triangle
The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by:
Example:
Find the area of the triangle with vertices A(1, 2), B(4, 5), and C(6, 3).
Solution: Area = ½ |1(5-3) + 4(3-2) + 6(2-5)|
= ½ |1×2 + 4×1 + 6×(-3)|
= ½ |2 + 4 - 18| = ½ |-12| = 6 square units
Visual representation of coordinate geometry concepts
Practice Questions
Question 1
What is the distance between points P(3, 4) and Q(7, 1)?
Answer: C) 5 units
Explanation: Using distance formula:
d = √[(7-3)² + (1-4)²] = √[4² + (-3)²] = √[16 + 9] = √25 = 5 units
Question 2
In which quadrant does the point (-3, 5) lie?
Answer: B) Quadrant II
Explanation: In Quadrant II, x-coordinate is negative and y-coordinate is positive.
Question 3
Find the midpoint of the line segment joining A(2, 3) and B(6, 9).
Answer: A) (4, 6)
Explanation: Using midpoint formula:
x = (2+6)/2 = 4, y = (3+9)/2 = 6
Midpoint: (4, 6)
Question 4
What is the area of the triangle with vertices (0, 0), (4, 0), and (0, 3)?
Answer: A) 6 square units
Explanation: Using area formula:
Area = ½ |0(0-3) + 4(3-0) + 0(0-0)| = ½ |0 + 12 + 0| = ½ × 12 = 6 square units
Question 5
The point which divides the line segment joining (2, 3) and (5, 6) in the ratio 2:1 is:
Answer: B) (4, 5)
Explanation: Using section formula:
x = (2×5 + 1×2)/(2+1) = (10+2)/3 = 12/3 = 4
y = (2×6 + 1×3)/(2+1) = (12+3)/3 = 15/3 = 5
Coordinates: (4, 5)
Question 6
The distance of the point (3, 4) from the origin is:
Answer: C) 5 units
Explanation: Using distance formula from origin (0,0):
d = √[(3-0)² + (4-0)²] = √[9 + 16] = √25 = 5 units
More Practice Problems
Problem 1
Show that the points (1, 1), (4, 4), and (6, 2) form a right-angled triangle.
Solution:
Let A(1, 1), B(4, 4), C(6, 2)
AB = √[(4-1)² + (4-1)²] = √[9+9] = √18
BC = √[(6-4)² + (2-4)²] = √[4+4] = √8
AC = √[(6-1)² + (2-1)²] = √[25+1] = √26
Now, (AB)² + (BC)² = 18 + 8 = 26 = (AC)²
Since Pythagoras theorem is satisfied, it's a right-angled triangle.
Problem 2
Find the coordinates of the point which divides the line segment joining (-3, 2) and (5, -4) in the ratio 3:2 externally.
Solution:
For external division, the formula is:
x = (mx₂ - nx₁)/(m - n), y = (my₂ - ny₁)/(m - n)
Here m:n = 3:2, (x₁,y₁)=(-3,2), (x₂,y₂)=(5,-4)
x = (3×5 - 2×(-3))/(3-2) = (15 + 6)/1 = 21
y = (3×(-4) - 2×2)/(3-2) = (-12 - 4)/1 = -16
Coordinates: (21, -16)
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