Hard

Dividing a Segment in a Ratio

Simple Explanation

Besides finding the exact middle of a segment, you can also find a point that divides it unevenly — for example, a point one-quarter of the way from one end to the other. This is called dividing a segment in a given ratio.

Why Do We Need It?

Not every "in-between" point is exactly in the middle — ratios let you locate any point along a segment precisely, which shows up in scaling, animation, and construction.

Worked Example

Divide a segment in a 1:3 ratio

Find the point that divides the segment from A(0, 0) to B(8, 12) in the ratio 1:3.

Why Does This Work?

Moving the same fraction of the way along each direction (x and y) independently moves you that same fraction of the way along the whole segment — the same idea as the midpoint, just with a fraction other than one-half.

Real-Life Example

Frame-by-frame animation

Animators move a character smoothly from a start position to an end position over several frames.

Each in-between frame places the character at a specific ratio along the segment from start to end — one-quarter of the way, half-way, and so on — using exactly this idea.

Practice

A point divides the segment from (0, 0) to (10, 20) in the ratio 1:1. Where is it?

Medium

Common mistake

Thinking a "1:3" ratio means "multiply by one-third" — it actually means the segment splits into 1+3=4 equal parts, so the fraction from A is 1/4, not 1/3.

Quick Review

Dividing a segment 'in a ratio' locates a point between two endpoints, not necessarily in the middle.
A ratio of m:n splits the segment into m+n equal parts.
The midpoint is just the special case of ratio 1:1.