Here is the completed worksheet with all blanks filled in, along with explanations 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 (or segments) intersect at a single location, that location is called a “point of concurrency.” All four triangle centers are defined by the intersection of three specific segments.
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CHART:
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1. Centroid
-
Special Segment: Medians
-
Important Facts:
- Always
inside the triangle
- The triangle’s
center of gravity (or
balance point)
>
✔ *Explanation:* The centroid is where the three medians (lines from each vertex to the midpoint of the opposite side) intersect. It’s always inside the triangle and acts as the triangle’s balance point — if you made a cardboard triangle, it would balance perfectly on a pin placed at the centroid.
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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 the intersection of the perpendicular bisectors of the sides. 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 where the three angle bisectors (lines that split each angle into two equal parts) meet. It’s always inside the triangle and is equidistant from all three sides — making it the center of the circle that fits snugly 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 (specifically, at the vertex of the right angle)
-
Outside an obtuse triangle
>
✔ *Explanation:* The orthocenter is the intersection point of the three altitudes (perpendicular lines from each vertex to the opposite side). Like the circumcenter, its position depends on the triangle type:
> - Acute → inside
> - Right → at the right-angle vertex
> - Obtuse → outside
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✔ Final Completed Worksheet:
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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.
| NAME OF CENTER | SPECIAL SEGMENT | PICTURE | IMPORTANT FACTS |
|----------------|--------------------------|---------|------------------|
|
Centroid | Medians | [Image] | • Always
inside the triangle<br>• The triangle’s
center of gravity |
|
Circumcenter | Perpendicular Bisectors | [Image] | •
Inside an acute triangle<br>•
On a right triangle<br>•
Outside an obtuse triangle<br>• Center of a
circumscribed circle |
|
Incenter | Angle Bisectors | [Image] | • Always
inside the triangle<br>• Center of an
inscribed circle |
|
Orthocenter| Altitudes | [Image] | •
Inside an acute triangle<br>•
On a right triangle<br>•
Outside an obtuse triangle |
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Let me know if you’d like diagrams or mnemonic devices to remember these!
Parent Tip: Review the logic above to help your child master the concept of centers of triangles worksheet.
