Directed Segments and the Section Formula

Lesson 1 of 2: Foundation

In this lesson:

• Understand directed segments and ratio notation
• Derive and apply the section formula
• Recognize the midpoint as a special case

Learning Objectives

By the end of this lesson, you should be able to:

1. Explain what a directed line segment is and how direction determines endpoint order
2. Derive and apply the section formula to find the point dividing a segment in ratio
3. Recognize the midpoint formula as the special case when the ratio is

You Know the Midpoint — Now Go Further

This gives the point exactly halfway from to .

Today's question: where is the point of the way? of the way? Any ratio?

Partitioning on a Number Line

• Number line from 0 to 10: of the way from 0 to 10 is 2.5
• In ratio notation: 2.5 divides [0, 10] in ratio 1:3
  • 1 part from 0 to 2.5; 3 parts from 2.5 to 10; total 4 parts
• of the way → ratio 3:1; the point is 7.5

Ratio — What It Means

In ratio from point to point :

• = number of parts from to
• = number of parts from to
• Total parts =
• Point is of the way from to

Example: ratio 2:3 → is of the way from to

Why Direction Determines the Partition Point

From to in ratio 1:3:

• is of the way from → closer to

From to in ratio 1:3:

• is of the way from → closer to

These are two different points on the same segment.

Always specify: partitioning from [starting point] to [ending point].

Partition Points on the Coordinate Plane

From to : midpoint at ; ratio 1:3 near ; ratio 3:1 near

Quick Check: Ratio Reasoning

In ratio 2:3 from to :

1. Is closer to or to ?
2. What fraction of equals ?

Think through both answers before the next slide.

Answer: Ratio 2:3 Means Two-Fifths

1. is closer to — the -side has fewer parts (2 vs. 3)

2. of — 2 parts out of 5 total

Key rule: ratio → fraction , not

Common error: treating ratio 2:3 as of the way — it is

Deriving the Section Formula Intuitively

Intuitive form: is of the way from to

• Start at
• = total horizontal distance
• Multiply by the fraction of the way you travel

Same logic gives:

Similar Triangles Justify the Section Formula

Similar triangles: divides each leg in ratio , so coordinates follow proportionally

Section Formula as a Weighted Average

Key: coefficient goes with (the target endpoint )

Quick Check: Which Coefficient Multiplies Endpoint B?

In the formula :

Which coefficient multiplies the target endpoint 's coordinate?

Name it and explain why before advancing.

Answer: m Multiplies Target Endpoint B

• multiplies (target endpoint ) ✓
• multiplies (starting endpoint )

Memory cue: " points to where you're going — "

Writing gives the wrong point — it reverses the result

Midpoint Is a Special Case

This is exactly the midpoint formula ✓

Any equal ratio (, ) gives the same result — all reduce to .

Worked Example: Find P in Ratio 2:1

Find on the segment from to in ratio .

Set up: , ; goes with 's coordinates

Worked Example 2 with Verification

to , ratio → find , then verify.

Verify: : , ; : ,

Ratio of -changes: ✓    Ratio of -changes: ✓

Your Turn: Apply the Section Formula

Find the point that divides to in ratio .

Step 1: Identify , , and which goes with which endpoint

Step 2: Set up and using

Step 3: Compute — expect a non-integer result for

Work all three steps before advancing.

Answer: Divides to in Ratio 1:4

, ; goes with

— fractional is correct; sits between grid lines

Verify: : , ; : , ;   ✓

Practice: Apply the Section Formula

Find on the segment from to in ratio .

Work all steps — set up, compute, verify — then advance.

Answer: to , Ratio 3:2

, ; goes with

Verify: : ; : ;   ✓

Key Takeaways

✓ Directed segment: direction sets which endpoint is start ()

✓ Ratio : is of the way from to

✓ Section formula: — multiplies target

✓ Midpoint: special case ; reduces to

Ratio → of the way — not

multiplies (target ) — not

Always name the start — direction changes the answer

Upcoming Topics and Curriculum Connections

Lesson 2 covers:

• Why the formula works — proof using similar triangles
• Four problem phrasings: ratio, fraction, percentage, reverse
• Real-world partition contexts

Connection backward: The section formula derives from similar triangles (HSG.SRT.A.2) — the same proportional reasoning you used for triangle similarity.

Connection forward: Partition points appear in coordinate proofs of medians and centroids (HSG.GPE.B.4).