Question

Point Y is the circumcenter of triangle DEF

Answer 1

Statement 1 can't be true, as if Y was the center, then it wouldn't be possible for a circle to pass through all 3 points; D, F and N.

Statement 2 can't be true, for the same reason as statement 1. If Y was the center, it wouldn't be possible for a circle to pass through all 3 points; M, N and E.

Statement 3 can't be true, for the same reasons. If Y was the center, it wouldn't be possible for a circle to pass through all 3 points; L, M and N.

Statement 4 is the true answer, as it is possible for a circle to pass through all 3 points; D, E and F if Y was the center. Also, Y is the circumcenter of triangle DEF, so this statement must be true.

Answer 2

Answer:

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

Step-by-step explanation:

Circumcenter-

1. The circumcenter is the center of the triangle's circumcircle and circumcircle is the circle that passes through all three of the triangle's vertices.
2. Circumcenter is the point where the perpendicular bisectors of a triangle intersect.

As point Y is the circumcenter of triangle DEF, so the circle, which is the circumcircle, with centre as Y must passes through the vertices D, E, F.