Coordinate Geometry Solved Examples (Class 10 Maths)
Coordinate geometry combines algebra and geometry using a coordinate system. It helps find distances between points, slopes of lines, and equations of line
TL;DR: Coordinate geometry combines algebra and geometry using a coordinate system. It helps find distances between points, slopes of lines, and equations of…
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Coordinate geometry combines algebra and geometry using a coordinate system. It helps find distances between points, slopes of lines, and equations of line
Coordinate Geometry — Solved Numerical Examples (Step by Step)
Example 1: Find the distance between points A(2, 3) and B(5, 7).
Solution: Given:
Point A: (x1, y1) = (2, 3)
Point B: (x2, y2) = (5, 7)
Using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((5 - 2)^2 + (7 - 3)^2)
d = sqrt((3)^2 + (4)^2)
d = sqrt(9 + 16)
d = sqrt(25)
d = 5
Example 2: Find the midpoint of the line segment joining points P(4, 6) and Q(8, 10).
Solution: Given:
Point P: (x1, y1) = (4, 6)
Point Q: (x2, y2) = (8, 10)
Using the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
M = ((4 + 8)/2, (6 + 10)/2)
M = (12/2, 16/2)
M = (6, 8)
Example 3: Find the slope of the line passing through points A(1, 2) and B(4, 8).
Solution: Given:
Point A: (x1, y1) = (1, 2)
Point B: (x2, y2) = (4, 8)
Using the slope formula: m = (y2 - y1) / (x2 - x1)
m = (8 - 2) / (4 - 1)
m = 6 / 3
m = 2
Example 4: Find the equation of the line passing through point (2, 5) with slope 3.
Solution: Given:
Point: (x1, y1) = (2, 5)
Slope m = 3
Using the point-slope form: y - y1 = m(x - x1)
y - 5 = 3(x - 2)
y - 5 = 3x - 6
y = 3x - 6 + 5
y = 3x - 1
Example 5: Check if the points A(1, 2), B(3, 4), and C(5, 6) are collinear.
Solution: Three points are collinear if the slope between A and B equals the slope between B and C.
Slope of AB: m1 = (4 - 2) / (3 - 1) = 2 / 2 = 1
Slope of BC: m2 = (6 - 4) / (5 - 3) = 2 / 2 = 1
Since m1 = m2 = 1, the points are collinear.
Example 6: Find the area of the triangle with vertices A(0, 0), B(4, 0), and C(2, 3).
Solution: Given:
Vertices: A(0, 0), B(4, 0), C(2, 3)
Using the triangle area formula:
Area = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Area = (1/2) |0(0 - 3) + 4(3 - 0) + 2(0 - 0)|
Area = (1/2) |0 + 12 + 0|
Area = (1/2) × 12
Area = 6 square units
Example 7: Find the equation of the line passing through points P(1, 1) and Q(2, 3).
Solution: First find the slope:
m = (3 - 1) / (2 - 1) = 2 / 1 = 2
Using the point-slope form with point P(1, 1):
y - 1 = 2(x - 1)
y - 1 = 2x - 2
y = 2x - 2 + 1
y = 2x - 1
Tips
- The distance formula is derived from the Pythagorean theorem. Always take the square root of the sum of squared differences in coordinates.
- The slope tells you the steepness of the line. If two lines have equal slopes, they are parallel. If the product of slopes is -1, the lines are perpendicular.
- For finding the area of a triangle using coordinates, use the formula with absolute value to ensure the area is positive. The order of vertices does not matter.
Frequently Asked Questions
What does a negative slope mean?
A negative slope means the line goes downward from left to right. For every unit increase in x, y decreases by the magnitude of the slope. For example, a slope of -2 means for every 1 unit right, the line goes 2 units down.
How do I find if two lines are perpendicular using slopes?
Two lines are perpendicular if the product of their slopes equals -1. For example, if one line has slope 2, a perpendicular line has slope -1/2. This is because perpendicular lines make 90-degree angles.
More Maths Solved Examples
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- Probability
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