Distance & SectionFoundation

Concept

Three workhorse formulas of coordinate geometry: the distance between two points, the midpoint halfway between them, and the section formula for the point dividing a segment in a given ratio $m:n$.

Key formula

Derivation

Distance comes straight from Pythagoras on the right triangle with legs $\Delta x$ and $\Delta y$.

For the section point dividing $AB$ internally as $m:n$, similar triangles give each coordinate as the weighted average $\dfrac{m\,(\text{far}) + n\,(\text{near})}{m+n}$.

The midpoint is the special case $m = n = 1$, averaging the coordinates.

Scenarios to explore

• Distance & Section — Distance, midpoint and section formula on the plane.

Real-world applications

• Centroid of a triangle as the average of its vertices.
• Dividing a line for graphics, surveying and interpolation.
• Checking collinearity and computing perimeters.

JEE exam tips

• External division uses $m:-n$ (a sign flip on $n$).
• Centroid $= \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$.

Common mistakes

• Swapping which point is weighted by $m$ vs $n$ in the section formula.
• Forgetting to square the differences before adding in the distance formula.
• Using the internal formula when the point divides the segment externally.

Exam traps to avoid

• The ratio is directed: $m:n$ from A means $A$ gets weight $n$, $B$ gets weight $m$.
• Distance is always non-negative; a negative under the root signals an arithmetic slip.