Partitioning a Segment
What if you don't want the halfway point? Learn to divide a segment in any given ratio.
Introduction
The midpoint splits a segment 1:1. But sometimes you need to split it 2:3, or 1:4, or any other ratio. The Section Formula (directed partition) handles all of these.
Past Knowledge
Midpoint on coordinate plane (1.2.3). Distance formula (1.2.4).
Today's Goal
Find a point that partitions a segment in a given ratio .
Future Success
Centroids of triangles (Unit 6) and dilations (Unit 7) use ratios on segments.
Key Concepts
The Section (Partition) Formula
To find the point that divides from to in the ratio :
Special Case: Midpoint
When , the fraction is , and the formula reduces to the midpoint formula.
Worked Examples
Ratio 1:3
Find the point that partitions from to in the ratio .
Fraction:
Answer:
Ratio 2:3
Find that partitions from to in .
Fraction:
Answer:
Common Pitfalls
Swapping the Ratio Direction
The ratio is from A to B. Ratio 1:3 means is of the way from A, not from B.
Using the Wrong Fraction
The fraction is , not .
Real-Life Applications
Animation & Game Dev
Game engines use linear interpolation (lerp) to move objects smoothly between two positions. Lerp is the Section Formula: at time , the position is — exactly our ratio formula with .
Practice Quiz
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