How do you derive the distance formula?

The distance formula comes from Pythagoras' theorem. For two points A(x₁,y₁) and B(x₂,y₂), draw a right triangle: horizontal leg = |x₂ − x₁|, vertical leg = |y₂ − y₁|. By Pythagoras: AB² = (x₂−x₁)² + (y₂−y₁)², so AB = √[(x₂−x₁)² + (y₂−y₁)²].

What is the section formula?

The section formula gives the coordinates of a point P that divides line segment AB in ratio m:n internally. If A = (x₁,y₁) and B = (x₂,y₂), then P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). Special case: when m = n (midpoint), the formula simplifies to ((x₁+x₂)/2, (y₁+y₂)/2).

How do you check if three points are collinear?

Three points are collinear (lie on the same straight line) if the area of the triangle formed by them is zero. Formula: Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. If this equals 0, the points are collinear. Alternatively, check if the slope between each pair of points is equal.

Distance and Section Formula

Coordinate Geometry • Class 10 Mathematics • NCERT • CBSE

Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²]. Section formula (internal division): P(x,y) = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n)). Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2).

Key Formulas

Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
Section formula: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Area of triangle = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Frequently Asked Questions

How do you derive the distance formula?: The distance formula comes from Pythagoras' theorem. For two points A(x₁,y₁) and B(x₂,y₂), draw a right triangle: horizontal leg = |x₂ − x₁|, vertical leg = |y₂ − y₁|. By Pythagoras: AB² = (x₂−x₁)² + (y₂−y₁)², so AB = √[(x₂−x₁)² + (y₂−y₁)²].
What is the section formula?: The section formula gives the coordinates of a point P that divides line segment AB in ratio m:n internally. If A = (x₁,y₁) and B = (x₂,y₂), then P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). Special case: when m = n (midpoint), the formula simplifies to ((x₁+x₂)/2, (y₁+y₂)/2).
How do you check if three points are collinear?: Three points are collinear (lie on the same straight line) if the area of the triangle formed by them is zero. Formula: Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. If this equals 0, the points are collinear. Alternatively, check if the slope between each pair of points is equal.

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