Educational worksheet on triangle centers including circumcenter, incenter, centroid, and orthocenter with diagrams and fill-in-the-blank questions.

Let’s go step by step to fill in the blanks for each center of the triangle.

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1. Circumcenter

- Created By: The circumcenter is created by the perpendicular bisectors of the sides of the triangle. (You can see in the diagram that lines are drawn perpendicular to the sides and cutting them in half — those are perpendicular bisectors.)

- Important Facts:
“The circumcenter is equidistant from each vertex of the triangle.”
→ Why? Because it’s the center of the circle that passes through all three vertices (the circumcircle), so distance to each vertex is the radius.

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2. Incenter

- Created By: The incenter is created by the angle bisectors of the triangle. (In the diagram, you see arcs showing angles being split equally — those are angle bisectors.)

- Important Facts:
“The incenter is equidistant from each side of the triangle.”
→ Why? Because it’s the center of the circle that touches all three sides (the incircle), so the perpendicular distance to each side is the same (radius).

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3. Centroid

- Created By: The centroid is created by the medians of the triangle. (A median connects a vertex to the midpoint of the opposite side — in the diagram, you see marks showing equal segments on the sides, meaning midpoints.)

- Important Facts:
“A median is created by a vertex connected to the midpoint of the opposite side.”
→ This defines what a median is — and the centroid is where all three medians meet.

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4. Orthocenter

- Created By: The orthocenter is created by the altitudes of the triangle. (An altitude is a line from a vertex perpendicular to the opposite side — in the diagram, you see right-angle symbols at the base, showing perpendicularity.)

- Important Facts:
“An altitude is created by a vertex connected to the opposite side so that it is perpendicular to that side.”
→ That’s the definition of an altitude — and the orthocenter is where all three altitudes intersect.

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Final Answer:

Circumcenter
Created By: Perpendicular Bisectors
Important Facts: The circumcenter is equidistant from each vertex of the triangle.

Incenter
Created By: Angle Bisectors
Important Facts: The incenter is equidistant from each side of the triangle.

Centroid
Created By: Medians
Important Facts: A median is created by a vertex connected to the midpoint of the opposite side.

Orthocenter
Created By: Altitudes
Important Facts: An altitude is created by a vertex connected to the opposite side so that it is perpendicular to that side.