Geometry

Question Analysis

This question is centered around the Triangle Perpendicular Bisector Concurrence Theorem. The core objective is to determine the type of triangle in which the circumcenter lies outside the triangle.

Key Concept Explanation

1. Circumcenter: It is the point of intersection of the perpendicular bisectors of a triangle’s sides. The circumcenter is the center of the circumcircle that passes through all three vertices of the triangle.

2. Perpendicular bisectors: A perpendicular bisector of a side of a triangle is a line that is perpendicular to the side and passes through its mid - point.

Step-by-Step Solution

1. Visualize an acute triangle: In an acute triangle, the perpendicular bisectors intersect inside the triangle. So, the circumcenter lies inside the acute triangle.

2. Visualize a right triangle: In a right triangle, the perpendicular bisectors intersect at the mid - point of the hypotenuse. So, the circumcenter lies on the hypotenuse.

3. Visualize an obtuse triangle: In an obtuse triangle, the perpendicular bisectors of the sides intersect outside the triangle. Thus, the circumcenter lies outside the obtuse triangle.

Option Analysis

A. Obtuse: Correct. As explained above, the circumcenter of an obtuse triangle lies outside the triangle. ...