Why It Works and Varied Problem Types

Lesson 2 of 2: Proof and Application

In this lesson:

• Prove the section formula using similar triangles
• Solve partition problems from four different phrasings
• Handle reverse problems: find the ratio given the partition point

Lesson 2 Learning Objectives: Proof and Application

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

4. Solve problems requiring a point a given fraction of the way from one endpoint to another
5. Connect the partition formula to proportional reasoning and similarity — explain why it works

Setting Up the Similarity Argument

Given and ; partitions in ratio .

Draw a right triangle with:

• Horizontal leg from to : length
• Vertical leg from to : length
• Hypotenuse = segment

Drop a perpendicular from → creates a smaller right triangle inside the larger one.

Similar Triangles Prove the Formula

Scale factor : both legs of the small triangle equal that fraction of the large

• Shared angle at guarantees the triangles are similar
• Similarity forces both legs to scale by the same fraction

Reading the Proof Step by Step

Horizontal leg of small triangle (scale factor ):

Therefore: — exactly the section formula ✓

Same argument gives: ✓

Verify Both Methods Give the Same Answer

Verify with , , ratio (scale factor ).

Horizontal: large leg ; small leg ;

Vertical: large leg ; small leg ;

Formula: ✓    ✓

Quick Check: Find the Scale Factor

For ratio from to :

What is the scale factor of the smaller triangle to the larger triangle?

Recall: scale factor , where and come from the ratio.

Write the scale factor as a fraction before advancing.

Answer: Scale Factor for Ratio 3:1

Scale factor

The small triangle's legs are the length of the large triangle's legs.

is of the way from to — closer to .

Partition Problems Come in Four Phrasings

Type 1: Ratio Given Directly

Find the point dividing to in ratio .

, ; goes with

Type 2: Convert a Fraction to a Ratio

Find the point of the way from to .

Convert: → 3 parts from , 1 part to → ratio 3:1; ,

Converting Fractions and Percentages to Ratios

Type 3: Partition in a Real-World Context

Trailhead to summit ; rest stop of the way from .

Convert: → ratio ; , ; goes with

Quick Check: Fraction to Ratio Conversion

Find the point of the way from to .

First: convert fraction to ratio — recall: → ratio

Then: compute using

Write both steps before advancing.

Answer: Find P Two-Fifths from A to B

Ratio: → ratio ; , ; goes with

Type 4: Find the Ratio from Point P

on segment to — find .

Coordinate differences:

• : ,
• : ,

-ratio:    -ratio: ✓   

Fractional Coordinates and Internal Division

Fractional coordinates are normal — the formula gives precise results, not rounded ones

• Example: is valid; it lies between grid lines

Internal division: both and → lies strictly between and

• Fraction ; equals 0 or 1 → is an endpoint
• Fraction outside → external division (beyond the scope of this standard)

Key Takeaways from Both Lessons Combined

✓ Formula from similar triangles — both legs scale by

✓ Fraction → ratio before applying

✓ Reverse problems: coordinate differences give

Ratio → fraction , not

multiplies (target ) — always label before computing

Fractional coordinates are valid — never round

Where This Topic Leads: Connections and Extensions

This lesson connects to:

• HSG.GPE.B.4 — partition points locate medians and centroids in coordinate proofs
• HSG.GPE.B.7 — perpendicular distances use partition point coordinates
• Calculus / CS: linear interpolation is the section formula with fraction
• Physics: center of mass = partition point with masses as the weights and