Question

in a right triangle, the circumcenter is located at the __________ of the hypotenuse.
the incenter is always located __________ the triangle.
the circumcenter of an obtuse triangle is located __________ the triangle.
the incenter is the center of the __________ of a triangle.
the circumcenter is the center of the __________ of a triangle.
the __________ is the point where the medians of a triangle intersect.
the __________ is the point where the altitudes of a triangle intersect.

Explanation:

1. Recall the property of the circum - center of a right triangle. The circum - center is the mid - point of the hypotenuse because the hypotenuse is the diameter of the circum - circle.
2. The in - center is the center of the incircle and is always inside the triangle as it is the intersection of the angle bisectors.
3. For an obtuse triangle, the circum - center is outside the triangle since the perpendicular bisectors of the sides intersect outside.
4. The in - center is the center of the incircle which is tangent to the sides of the triangle.
5. The circum - center is the center of the circum - circle that passes through the three vertices of the triangle.
6. The centroid is the point of intersection of the medians of a triangle.
7. The ortho - center is the point of intersection of the altitudes of a triangle.

Answer:

1. mid - point
2. inside
3. outside
4. incircle
5. circum - circle
6. centroid
7. ortho - center