Master the Class 9 Madhyamik Mathematics Coordinate Geometry Formulas including Distance Formula, Equidistant Point, Collinearity of Points, and coordinate-based proofs of Isosceles Triangle, Equilateral Triangle, Right-Angled Triangle, Square, Rectangle, Rhombus, Parallelogram, and Kite. Each concept is explained with step-by-step solved examples for quick learning and exam preparation.
Chapter 04 – Co-ordinate Geometry – Distance Formula | Class 9 Flash Formula
Distance Formula
Let A(x₁, y₁) and B(x₂, y₂) be the two points.
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example:
Find the distance between the points A(2, 3) and B(8, 11).
Using the formula,
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
= √[(8 − 2)² + (11 − 3)²]
= √[6² + 8²]
= √[36 + 64]
= √100
= 10 units
Equidistant Point Formula
If a point P(x, y) is equidistant from two points A(x₁, y₁) and B(x₂, y₂), then
PA = PB
i.e.
√[(x − x₁)² + (y − y₁)²] = √[(x − x₂)² + (y − y₂)²]
After squaring both sides,
(x − x₁)² + (y − y₁)² = (x − x₂)² + (y − y₂)²
If P(x, 4) is equidistant from A(1, 2) and B(5, 6), find the value of x.
Since P is equidistant from A and B,
PA = PB
√[(x − 1)² + (4 − 2)²] = √[(x − 5)² + (4 − 6)²]
Squaring both sides,
(x − 1)² + 4 = (x − 5)² + 4
(x − 1)² = (x − 5)²
x² − 2x + 1 = x² − 10x + 25
8x = 24
x = 3
Therefore, P = (3, 4).
Condition of Collinearity
Three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are collinear if
Distance AB + Distance BC = Distance AC
Example (Using Distance Formula)
Check whether the points A(1, 2), B(3, 4) and C(5, 6) are collinear.
Distance AB:
AB = √[(3 − 1)² + (4 − 2)²] = 2√2
Distance BC:
BC = √[(5 − 3)² + (6 − 4)²] = 2√2
Distance AC:
AC = √[(5 − 1)² + (6 − 2)²] = 4√2
Now,
AB + BC = 2√2 + 2√2 = 4√2 = AC
Therefore, AB + BC = AC
Condition for Isosceles Triangle
A triangle is isosceles if any two sides are equal.
AB = AC or AB = BC or AC = BC
Example
Let A(0, 0), B(4, 0) and C(2, 3)
AB = √[(4 − 0)² + (0 − 0)²] = 4
AC = √[(2 − 0)² + (3 − 0)²] = √13
BC = √[(2 − 4)² + (3 − 0)²] = √13
Since AC = BC,
ΔABC is an Isosceles Triangle.
Condition for Right-Angled Triangle
A triangle is right-angled if
AB² + BC² = AC² (where AC is the longest side)
Example
Let A(0, 0), B(3, 0) and C(3, 4)
AB = 3, BC = 4 and AC = 5
AB² + BC² = 3² + 4² = 25
AC² = 5² = 25
Since AB² + BC² = AC²,
ΔABC is a Right-Angled Triangle.
Condition for Equilateral Triangle
A triangle is equilateral if
Length of AB = Length of BC = Length of AC
Example
Let A(0, 0), B(2, 0) and C(1, √3)
AB = √[(2 − 0)² + (0 − 0)²] = 2
AC = √[(1 − 0)² + (√3 − 0)²] = 2
BC = √[(1 − 2)² + (√3 − 0)²] = 2
Since AB = BC = AC = 2,
ΔABC is an Equilateral Triangle.
Condition for Square
A quadrilateral is a Sqaure if
Sides : Length of AB = Length of BC = Length of CD = Length of DA and
Diagonals: Length of AC = Length of BD
You have reached the end.