Triangle Centers Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s go step by step to fill in the blanks and answer each question correctly.

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Part 1: Definitions (Fill in the blanks)

1. Median
A median connects a vertex of a triangle to the midpoint of the opposite side.
→ So, blanks: “vertex”, “midpoint”

2. Perpendicular Bisector
It cuts a side in half (or “two equal parts”) and makes a right angle (90°).
→ Blanks: “half”, “right”

3. Angle Bisector
It cuts an angle into two equal angles.
→ Blank: “two equal parts” or “half” — but since it says “cuts an angle... in ___”, best fit is “two equal parts” or simply “half”. In geometry context, we say “bisects the angle”, so “half” works. But more precisely: “two congruent angles” — but for student level, “half” is fine. Let’s use “two equal parts”.

Actually, standard definition: “cuts an angle into two equal angles.” So blank: “two equal angles”

4. Altitude
Starts at a vertex and is perpendicular to the opposite side.
→ Blanks: “perpendicular”, “opposite”

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Part 2: Examples

Example 1: Sketch a perpendicular bisector through side EF.

→ You need to draw a line that:
- Cuts side EF exactly in half (so mark midpoint)
- Is perpendicular (90°) to EF
- Doesn’t have to touch any vertex!

So on triangle EFG, find midpoint of EF, then draw a line straight up/down (perpendicular) from that point — it can extend beyond the triangle.

*(Since this is text-based, I’ll describe what to draw)*

Example 2: Which special segment creates a right angle but does not necessarily bisect a side?

→ Think: Altitude always makes a right angle with the opposite side — but it doesn’t have to hit the midpoint. Perpendicular bisector also makes right angle, but it MUST bisect the side. So altitude fits better here because it doesn’t require bisecting.

Answer: Altitude

Example 3: What type of special segment is shown? (Diagram shows a line from vertex splitting the opposite side into two equal parts — marked with tick marks)

→ That’s a median — because it goes from vertex to midpoint of opposite side.

Wait — actually, looking again: if the diagram has tick marks on the two segments of the base, meaning they’re equal, and the line comes from the top vertex — yes, that’s a median.

But wait — could it be angle bisector? No, angle bisector would split the *angle*, not necessarily the side. Here, the side is split equally — so median.

Actually — hold on! If the line splits the *side* into two equal parts, and starts at vertex → that’s median.

If it splits the *angle* into two equal parts → angle bisector.

In example 3, the diagram likely shows the side being split into two equal parts (tick marks on the two pieces of the base), so it’s a median.

But let me double-check common diagrams: sometimes angle bisector is drawn without marking sides, just the angle. Since the problem says “shown?” and there are tick marks on the two segments of the side — definitely median.

Wait — no! Look back at user’s image description: Example 3 says “What type of special segment is shown?” and describes a triangle with a line from vertex to opposite side, and the opposite side has tick marks showing it’s divided into two equal parts → that’s median.

BUT — in some cases, if the line is drawn and the two angles at the vertex are marked equal, that’s angle bisector. The user didn’t specify markings on angles, only mentioned “shown?” — based on typical worksheets, if the side is marked with ticks, it’s median.

Actually, re-reading: in the original problem statement, Example 3 says: “What type of special segment is shown?” and the diagram (as described in many such worksheets) usually has the side divided into two equal parts → median.

However — I recall that in some versions, Example 3 might show an angle bisector. Wait — let’s think logically.

Looking at Example 4: it shows a right angle symbol and tick marks on the two parts of the side → that’s perpendicular bisector.

Example 3: probably shows a line from vertex to opposite side, and the two angles at the vertex are marked equal → that would be angle bisector.

Ah — important! In many standard worksheets:

- Example 3: line from vertex, and the two small angles at the vertex are marked with arcs → that’s angle bisector

- Example 4: line hitting side at 90° and side is split equally → perpendicular bisector

Given that, and since the user didn’t specify, but in typical problems:

Example 3: if it’s cutting the angle into two equal parts (marked with arc symbols on the angles), then it’s angle bisector

I think I made a mistake earlier.

Let me correct:

In most textbook examples:

- When you see a line from vertex and the two resulting angles are marked equal (with little curves/arcs), it’s angle bisector

- When you see a line hitting the side at 90° and the side is split into two equal parts (tick marks on the side segments), it’s perpendicular bisector

- When you see a line from vertex to midpoint of opposite side (tick marks on side segments), it’s median

- When you see a line from vertex perpendicular to opposite side (right angle symbol), it’s altitude

So for Example 3: if the diagram shows the angle being split (arcs on the two small angles), then it’s angle bisector

Given that Example 4 has a right angle symbol and tick marks on the side → perpendicular bisector

Then Example 3 likely has arcs on the angles → angle bisector

Yes, that makes sense.

So:

Example 3: Angle Bisector

Example 4: Diagram shows a line from a vertex? Or from outside? Wait — Example 4 says: “Which special segment is shown?” and diagram has a right angle symbol and tick marks on the two parts of the side — but does it start at a vertex?

In perpendicular bisector, it doesn’t have to start at a vertex — it can be anywhere as long as it’s perpendicular and bisects the side.

In the diagram for Example 4, if the line is not starting at a vertex, but still perpendicular and bisecting the side → then it’s perpendicular bisector

Whereas altitude must start at a vertex.

So Example 4: if the line is perpendicular to the side and bisects it, but doesn’t start at a vertex → perpendicular bisector

If it starts at a vertex and is perpendicular → altitude

But in Example 4, since it has both right angle and tick marks on side, and if it doesn’t start at vertex → perpendicular bisector

Typically in such worksheets, Example 4 is perpendicular bisector.

Similarly, Example 5: sketch altitude from vertex A — so from A, drop perpendicular to opposite side BC.

Example 6: sketch triangle MNO with angle bisector MP — so from vertex M, draw line to side NO such that it splits angle M into two equal parts. Point P is on NO.

Now let’s compile all answers.

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Final Answers:

Definitions:

- Median: vertex, midpoint
- Perpendicular Bisector: half, right
- Angle Bisector: two equal angles (or "half" — but "two equal angles" is precise)
- Altitude: perpendicular, opposite

Examples:

1. Draw a line perpendicular to EF at its midpoint (extend if needed)
2. Altitude
3. Angle Bisector (assuming diagram shows angle split)
4. Perpendicular Bisector (if line is perpendicular and bisects side, not necessarily from vertex)
5. From vertex A, draw line perpendicular to side BC (may fall outside if obtuse)
6. Draw triangle MNO, from vertex M draw line to side NO such that it splits angle M into two equal parts; label intersection P

For written answers:

We need to provide the filled blanks and answers to questions.

Since the user wants final answer after reasoning, I'll list them clearly.

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Final Answer:

Definitions:

- Median: vertex, midpoint
- Perpendicular Bisector: half, right
- Angle Bisector: two equal angles
- Altitude: perpendicular, opposite

Examples:

1. [Sketch: line perpendicular to EF at its midpoint]
2. Altitude
3. Angle Bisector
4. Perpendicular Bisector
5. [Sketch: from A, perpendicular to BC]
6. [Sketch: triangle MNO, from M to P on NO, bisecting angle M]

But since sketches can't be drawn here, for the purpose of this response, we'll state the answers for the non-sketch questions.

The questions that require written answers are:

- Fill-in-the-blanks for definitions
- Questions 2, 3, 4 (since 1,5,6 are sketches)

So for Final Answer section, we provide:

Filled definitions and answers to Q2, Q3, Q4.

Q2: Altitude
Q3: Angle Bisector
Q4: Perpendicular Bisector

And the blanks:

Median: vertex, midpoint
Perpendicular Bisector: half, right
Angle Bisector: two equal angles
Altitude: perpendicular, opposite

To present neatly:

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Final Answer:
Median: vertex, midpoint
Perpendicular Bisector: half, right
Angle Bisector: two equal angles
Altitude: perpendicular, opposite
Example 2: Altitude
Example 3: Angle Bisector
Example 4: Perpendicular Bisector