Question

point y is the circumcenter of triangle def. which statement is true about point y? point y is the center of the circle that passes through points d, f, and n. point y is the center of the circle that passes through points m, n, and e. point y is the center of the circle that passes through points l, m, and n. point y is the center of the circle that passes through points d, e, and f.

Explanation:

The circumcenter of a triangle is the point where the perpendicular bisectors of the triangle's sides intersect. By definition, the circumcenter is equidistant from all three vertices of the triangle, and it is the center of the circumscribed circle (circumcircle) that passes through all three vertices of the triangle.

For triangle \( DEF \), the circumcenter \( Y \) should be the center of the circle that passes through its three vertices \( D \), \( E \), and \( F \).

Let's analyze the other options:

• The first option mentions a circle passing through \( D \), \( F \), and \( N \). There's no reason to assume \( N \) is a vertex or related to the circumcircle of \( \triangle DEF \) in this way.
• The second option mentions a circle passing through \( M \), \( N \), and \( E \). \( M \) and \( N \) are points on the perpendicular bisectors, not vertices of the triangle, so this is incorrect.
• The third option mentions a circle passing through \( L \), \( M \), and \( N \). These are points on the perpendicular bisectors, not vertices, so this is also incorrect.

Answer:

D. Point Y is the center of the circle that passes through points D, E, and F.