Midsegment Theorem and Centroid Explained

Lesson 2 of 2

In this lesson:

• Prove the Triangle Midsegment Theorem (parallel and half-length)
• Prove that the three medians meet at the centroid (2:1 ratio)
• Apply both theorems with coordinates and calculations

Lesson Goals for Midsegment and Centroid

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

1. Prove the Midsegment Theorem: midpoint connector is parallel to the third side and half its length
2. Prove that three medians meet at the centroid; apply the 2:1 ratio
3. Solve problems using coordinates and both theorems

What Happens When You Connect Midpoints?

Draw . Mark:

• = midpoint of
• = midpoint of

Connect and .

Measure and . What ratio do you find?
Check: does look parallel to ?

This investigation suggests two properties — let's prove both.

Triangle Midsegment Theorem: Two Key Properties

If is the midpoint of and is the midpoint of , then and .

Key: connects midpoints of two sides — and is parallel to the third side.

Proof Setup: Using SAS Similarity

Given: is midpoint of , is midpoint of
To prove: and

Since and are midpoints:

So

And (shared angle — reflexive)

Midsegment Proof — Part 1: Establishing Similarity

Given: midpoint of , midpoint of , through

Midsegment Proof — Part 2: Drawing Both Conclusions

Continuing from (scale factor ):

Quick Check: Applying the Midsegment Theorem

In :

• is the midpoint of
• is the midpoint of
• cm

Find . Is ?

State the theorem, then compute.

Answer: Applying the Midsegment Theorem Here

By the Triangle Midsegment Theorem: ✓ and cm ✓

Coordinate check: For , , , midpoints and :

✓ · slopes equal ✓

Medial Triangle: Three Midsegments Together

The three midsegments of a triangle form the medial triangle.

• Similar to with scale factor
• Divides the original into four congruent triangles
• Each midsegment is half the opposite side's length

Coordinate Verification: Check the Midsegment

Triangle with vertices , , :

1. Find midpoints (of ) and (of )
2. Compute and — verify
3. Compute slopes of and — are they equal?

Work through all three steps before advancing.

Defining and Drawing Triangle Medians

A median connects a vertex to the midpoint of the opposite side. Every triangle has exactly three medians — one from each vertex.

The three medians appear to intersect at a single point.

Medians Are Concurrent: The Centroid

Theorem: The three medians meet at a single point — the centroid — dividing each median in a 2:1 ratio from vertex to midpoint.

Coordinate Proof — Part 1: Setup and Midpoints

Place , , . Compute midpoints:

Parametrize the median from ( at , at ):

Coordinate Proof — Part 2: Finding the Intersection

Parametrize the median from :

Set . From -coordinates: . From -coordinates:

Intersection: — the average of all three vertices.

Centroid Formula and Balance Point Interpretation

Balance point: A triangle balances on a pencil tip placed at .

2:1 ratio: — the centroid is of the way from each vertex to the opposite midpoint.

Quick Check: Finding the Centroid

Triangle with vertices , , .

Find the centroid using the formula.

Then verify: compute the midpoint of and confirm that is 2/3 of the way from to .

Try both steps before advancing.

Answer: Centroid Calculation and Ratio Verification

Centroid formula:

Midpoint of :

Verify 2:1 — point of the way from to :

Quick Check: The 2:1 Centroid Ratio

A median of a triangle has length 15 units.

1. How far is the centroid from the vertex?
2. How far is the centroid from the midpoint of the opposite side?

Name the theorem, state the ratio, then compute.

Answer: Centroid Is Two-Thirds from Vertex

Centroid Theorem: 2:1 ratio from vertex to midpoint.

• Vertex to centroid: units
• Centroid to midpoint: units

Centroid is 2/3 from vertex — NOT at the median midpoint.

Key Takeaways from Lesson Two

✓ Midsegment: midpoint connector ∥ third side, half its length (SAS)
✓ Centroid: three medians meet at ; 2:1 ratio from vertex
✓ Formula:

Midsegment ∥ third side — not the sides it connects
Centroid = 2/3 from vertex, not median midpoint

What Comes Next: Parallelogram Theorems

Next lesson: HSG.CO.C.11 — Parallelogram Theorems

Your triangle toolkit will reappear:

• Angle sum → parallelogram interior angles
• Congruence criteria → proving opposite sides and angles equal
• Midpoint results → diagonal properties

Every theorem you prove becomes a tool for the next lesson.