Question

(a) select all that describe \\(\overline{gi}\\).

• perpendicular bisector of \\(\overline{fh}\\)
• angle bisector of \\(\angle g\\)
• median of \\(\triangle fgh\\)
• altitude of \\(\triangle fgh\\)
• none of the above

(image of \\(\triangle fgh\\) with (g) at the top, (f) and (h) at the base, (i) on (fh))

(b) select all that describe \\(\overline{zv}\\).

• perpendicular bisector of \\(\overline{xy}\\)
• angle bisector of \\(\angle z\\)
• median of \\(\triangle xyz\\)
• altitude of \\(\triangle xyz\\)
• none of the above

(image of \\(\triangle xyz\\) with (z) at the right, (x) at the left, (y) at the top, (v) on (xy) with a right angle)

(c) select all that describe \\(\overline{ad}\\).

• perpendicular bisector of \\(\overline{bc}\\)
• angle bisector of \\(\angle a\\)
• median of \\(\triangle abc\\)
• altitude of \\(\triangle abc\\)
• none of the above

(image of \\(\triangle abc\\) with (a) at the left, (c) at the right, (b) at the top, (d) on (bc) with marks indicating equal segments)

Explanation:

• For part (a):
• A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two equal parts. If \( \overline{GI}\) is the perpendicular bisector of \( \overline{FH}\), it meets the definition.
• An angle bisector divides an angle into two equal angles. Since \( \overline{GI}\) divides \( \angle G\) into two equal angles, it is an angle bisector.
• A median of a triangle is a line segment joining a vertex to the mid - point of the opposite side. If \(I\) is the mid - point of \( \overline{FH}\), then \( \overline{GI}\) is a median of \( \triangle FGH\).
• An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. If \( \overline{GI}\) is perpendicular to \( \overline{FH}\), it is an altitude.
• For part (b):
• A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two equal parts. If \( \overline{ZV}\) is the perpendicular bisector of \( \overline{XY}\), it meets the definition.
• An angle bisector divides an angle into two equal angles. Since \( \overline{ZV}\) divides \( \angle Z\) into two equal angles, it is an angle bisector.
• A median of a triangle is a line segment joining a vertex to the mid - point of the opposite side. If \(V\) is the mid - point of \( \overline{XY}\), then \( \overline{ZV}\) is a median of \( \triangle XYZ\).
• An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Since \( \overline{ZV}\) is perpendicular to \( \overline{XY}\), it is an altitude.
• For part (c):
• A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two equal parts. There is no indication that \( \overline{AD}\) is perpendicular to \( \overline{BC}\), so it is not a perpendicular bisector.
• An angle bisector divides an angle into two equal angles. There is no indication that \( \overline{AD}\) divides \( \angle A\) into two equal angles, so it is not an angle bisector.
• A median of a triangle is a line segment joining a vertex to the mid - point of the opposite side. \(D\) is the mid - point of \( \overline{BC}\), so \( \overline{AD}\) is a median of \( \triangle ABC\).
• An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. There is no indication that \( \overline{AD}\) is perpendicular to \( \overline{BC}\), so it is not an altitude.

Answer:

(a) Perpendicular bisector of \( \overline{FH}\), Angle bisector of \( \angle G\), Median of \( \triangle FGH\), Altitude of \( \triangle FGH\)
(b) Perpendicular bisector of \( \overline{XY}\), Angle bisector of \( \angle Z\), Median of \( \triangle XYZ\), Altitude of \( \triangle XYZ\)
(c) Median of \( \triangle ABC\)