### Mathematics Class X

Real Numbers
Polynomials
Arithmetic Progressions
Triangles
Coordinate Geometry
Introduction to Trigonometry
Circles
Constructions
Areas Related to Circles
Surface Areas and Volumes
Statistics
Probability

# Collinearity Condition

If three points A, B and C are collinear and B lies between A and C, then,

• AB + BC = AC.  AB, BC, and AC can be calculated using the distance formula.
• The ratio in which B divides AC, calculated using section formula for both the x and y coordinates separately will be equal.
• Area of a triangle formed by three collinear points is zero.

NOTES

(I) Four points will form:

(a) A parallelogram if it’s opposite sides is equal, but diagonals are unequal.

(b) A rectangle if opposite sides is equal and two diagonals are also equal.

(c) A rhombus if all the four sides are equal, but diagonals unequal,

(d) A square if all sides are equal and diagonals are also equal.

(II) Three points will form:

(a) An equilateral triangle if all the three sides are equal.

(b) An isosceles triangle if any two sides are equal.

(c) A right angled triangle if sum of square of any two sides is equal to square of the third side.

(d) A triangle if sum of any two sides (distances) is greater than the third side (distance).

(III) Three points A, B and C are collinear or lie on a line if one of the following holds

(i) AB + BC — AC

(ii) AC + CB AB

(iii) CA + AB CB.

Example
Find the distance of the point (-3, 4) from the x-axis.
Solution:
B(-3, 0), A (-3, 4)  Example.
If the points A(x, 2), B(-3, 4) and C(7, -5) are collinear, then find the value of x.
Solution:
When the points are collinear, Example
In which quadrant the point P that divides the line segment joining the points A (2, -5) and B(5,2) in the ratio 2 : 3 lies?
Solution:   IV Quadrant

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