Question

which types of centers are always inside the triangle? (select all that apply) circumcenter incenter centroid orthocenter

Explanation:

• Incenter: The incenter is the intersection of the angle bisectors of a triangle. It is always inside the triangle as it is equidistant from all sides of the triangle.
• Centroid: The centroid is the intersection of the medians of a triangle. Medians connect a vertex to the mid - point of the opposite side. Since mid - points of sides are on the triangle (not extended), the centroid is always inside the triangle.
• Circumcenter: In an acute triangle, the circumcenter (intersection of perpendicular bisectors) is inside. In a right - triangle, it is at the mid - point of the hypotenuse. In an obtuse triangle, it is outside. So, it is not always inside.
• Orthocenter: In an acute triangle, the orthocenter (intersection of altitudes) is inside. In a right - triangle, it is at the right - angled vertex. In an obtuse triangle, it is outside. So, it is not always inside.

Answer:

Incenter, Centroid