Triangle Centers Notes and Worksheets - Lindsay Bowden - Free Printable

Here is the completed worksheet with all blanks filled in, along with a clear explanation for each answer.

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TRIANGLE CENTERS notes

• Triangle Centers – When 3 special segments in a triangle meet at a point, they create a triangle center or a point of concurrency. The 4 triangle centers are in the chart below.

> ✔ *Explanation*: In geometry, when three lines (like medians, altitudes, etc.) intersect at a single location inside or on a triangle, that location is called a “point of concurrency.” So the blanks are filled with: point, point, and concurrency.

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CHART COMPLETION:

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1. Centroid
- Special Segment: Medians
- Important Facts:
- Always inside the triangle
- The triangle’s center of mass or balance point

> ✔ *Explanation*: The centroid is formed by the intersection of the three medians (lines from each vertex to the midpoint of the opposite side). It is always located inside the triangle and acts as the triangle’s center of gravity — if you made a cardboard triangle, it would balance perfectly on this point.

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2. Circumcenter
- Special Segment: Perpendicular Bisectors
- Important Facts:
- Inside an acute triangle
- On a right triangle *(specifically, at the midpoint of the hypotenuse)*
- Outside an obtuse triangle
- Center of a circumscribed circle *(also called the circumcircle)*

> ✔ *Explanation*: The circumcenter is where the perpendicular bisectors of the sides meet. It is equidistant from all three vertices, so it’s the center of the circle that passes through all three vertices (the circumcircle). Its location depends on the triangle type:
> - Acute → inside
> - Right → on the hypotenuse (midpoint)
> - Obtuse → outside

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3. Incenter
- Special Segment: Angle Bisectors
- Important Facts:
- Always inside the triangle
- Center of an inscribed circle *(also called the incircle)*

> ✔ *Explanation*: The incenter is formed by the intersection of the angle bisectors. It is always inside the triangle and is equidistant from all three sides. This makes it the center of the circle that fits perfectly inside the triangle and touches all three sides (the incircle).

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4. Orthocenter
- Special Segment: Altitudes
- Important Facts:
- Inside an acute triangle
- On a right triangle *(at the right-angle vertex)*
- Outside an obtuse triangle

> ✔ *Explanation*: The orthocenter is where the three altitudes (perpendicular lines from each vertex to the opposite side) intersect. Like the circumcenter, its location varies:
> - Acute → inside
> - Right → exactly at the vertex of the right angle
> - Obtuse → outside the triangle

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✔ Final Completed Worksheet Summary:

| NAME OF CENTER | SPECIAL SEGMENT | PICTURE | IMPORTANT FACTS |
|----------------|------------------------|---------|------------------|
| Centroid | Medians | | • Always inside the triangle<br>• The triangle’s center of mass |
| Circumcenter | Perpendicular Bisectors| | • Inside an acute triangle<br>• On a right triangle<br>• Outside an obtuse triangle<br>• Center of a circumscribed circle |
| Incenter | Angle Bisectors | | • Always inside the triangle<br>• Center of an inscribed circle |
| Orthocenter | Altitudes | | • Inside an acute triangle<br>• On a right triangle<br>• Outside an obtuse triangle |

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📌 Bonus Tip: Remember these four centers using the acronym CICO:
- Centroid → Medians → Balance point
- Incenter → Angle bisectors → Inscribed circle
- Circumcenter → Perp. bisectors → Circumscribed circle
- Orthocenter → Altitudes → Intersection of heights

Let me know if you’d like diagrams or flashcards to help memorize them!