Coordinate Geometry Solved Examples (Class 10 Mathematics)
Calculate distances, find midpoints, and analyze geometric figures using coordinate systems. These problems connect algebra with geometry.
TL;DR: Calculate distances, find midpoints, and analyze geometric figures using coordinate systems. These problems connect algebra with geometry.
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Calculate distances, find midpoints, and analyze geometric figures using coordinate systems. These problems connect algebra with geometry.
Coordinate Geometry — Solved Numerical Examples (Step by Step)
Example 1: Find the distance between points A(3, 4) and B(6, 8).
Solution: Using distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
d = √((6 - 3)² + (8 - 4)²) = √(3² + 4²) = √(9 + 16) = √25 = 5
Example 2: Find the midpoint of the line segment joining A(2, 3) and B(6, 7).
Solution: Using midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
M = ((2 + 6)/2, (3 + 7)/2) = (8/2, 10/2) = (4, 5)
Example 3: Verify that the points A(0, 0), B(3, 4), and C(0, 5) form a right triangle. (Hint: Check if one angle is 90°)
Solution: Calculate distances:
AB = √((3-0)² + (4-0)²) = √(9 + 16) = √25 = 5
BC = √((0-3)² + (5-4)²) = √(9 + 1) = √10
AC = √((0-0)² + (5-0)²) = √25 = 5
Check Pythagorean theorem: AB² + AC² = BC²?
25 + 25 = 50, but BC² = 10 (not matching)
Let's check: AB² + BC² = AC²?
25 + 10 = 35, but AC² = 25 (not matching)
Let's check: AC² + BC² = AB²?
25 + 10 = 35, but AB² = 25 (not matching)
Actually, none of these work. The triangle is NOT a right triangle with these distances.
Example 4: Find the equation of the line passing through points P(2, 3) and Q(4, 7).
Solution: Slope m = (y₂ - y₁) / (x₂ - x₁) = (7 - 3) / (4 - 2) = 4/2 = 2
Using point-slope form: y - y₁ = m(x - x₁)
y - 3 = 2(x - 2)
y - 3 = 2x - 4
y = 2x - 1
Example 5: The distance between A(x, 3) and B(2, 1) is 2√5. Find the value of x.
Solution: Using distance formula: √((2 - x)² + (1 - 3)²) = 2√5
√((2 - x)² + 4) = 2√5
Squaring both sides: (2 - x)² + 4 = 20
(2 - x)² = 16
2 - x = ±4
If 2 - x = 4, then x = -2
If 2 - x = -4, then x = 6
So x = -2 or x = 6
Example 6: Show that the quadrilateral with vertices A(0, 0), B(4, 0), C(4, 3), and D(0, 3) is a rectangle.
Solution: For a rectangle, opposite sides must be equal and all angles 90°
AB = √((4-0)² + (0-0)²) = 4
CD = √((0-4)² + (3-3)²) = 4 (opposite sides equal)
BC = √((4-4)² + (3-0)²) = 3
AD = √((0-0)² + (3-0)²) = 3 (opposite sides equal)
Diagonal AC = √((4-0)² + (3-0)²) = √(16 + 9) = 5
Diagonal BD = √((0-4)² + (3-0)²) = √(16 + 9) = 5 (diagonals equal)
Since opposite sides are equal and diagonals are equal, ABCD is a rectangle.
Tips
- Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²); memorize this well.
- Midpoint divides a segment in 1:1 ratio; for section formula with ratio m:n, point = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).
- Slope m = (y₂ - y₁)/(x₂ - x₁); perpendicular lines have slopes m₁ × m₂ = -1.
Frequently Asked Questions
How do you find if three points are collinear?
Three points are collinear if the area of the triangle they form is zero, or if the slope between any two pairs is equal.
What is the relationship between the slopes of perpendicular lines?
If two lines are perpendicular, the product of their slopes is -1 (m₁ × m₂ = -1). For example, slopes 2 and -1/2 represent perpendicular lines.
More Mathematics Solved Examples
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