Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.) answer attempt 1 out of 2 $overline{hi}$ is a segment bisector. $overline{hi}$ is a perpendicular bisector. $h$ is the vertex of a right angle. $i$ is the vertex of a right angle. $h$ is the midpoint of a segment in the diagram. $i$ is the midpoint of a segment in the diagram

Explanation:

1. For \(\overline{HI}\) being a segment bisector: A segment bisector divides a segment into two equal parts. From the diagram, \(I\) divides \(GF\) into two equal parts (marked with two ticks) and \(H\) divides \(GE\) into two equal parts (marked with one tick), and \(\overline{HI}\) connects these midpoints, so it bisects segments. So this is true.
2. For \(\overline{HI}\) being a perpendicular bisector: A perpendicular bisector is a segment bisector that is also perpendicular to the segment. Here, \(\overline{HI}\) is perpendicular to \(\overline{GE}\) (right angle at \(H\)) and \(H\) is the midpoint of \(\overline{GE}\), so \(\overline{HI}\) is a perpendicular bisector of \(\overline{GE}\). Also, \(I\) is the midpoint of \(\overline{GF}\), and if we consider the other segment, but the right angle at \(H\) shows perpendicularity. So this is true.
3. For \(H\) being the vertex of a right angle: The diagram shows a right angle symbol at \(H\), so \(H\) is the vertex of a right angle. This is true.
4. For \(I\) being the vertex of a right angle: There is no right angle symbol at \(I\), so this is false.
5. For \(H\) being the midpoint of a segment: The marks on \(\overline{GE}\) show that \(H\) divides \(\overline{GE}\) into two equal parts, so \(H\) is the midpoint of \(\overline{GE}\). This is true.
6. For \(I\) being the midpoint of a segment: The marks on \(\overline{GF}\) show that \(I\) divides \(\overline{GF}\) into two equal parts, so \(I\) is the midpoint of \(\overline{GF}\). This is true (assuming the last option was a typo and meant midpoint of a segment, as per the diagram).

Answer:

• \(\boldsymbol{\overline{HI}}\) is a segment bisector.
• \(\boldsymbol{\overline{HI}}\) is a perpendicular bisector.
• \(\boldsymbol{H}\) is the vertex of a right angle.
• \(\boldsymbol{H}\) is the midpoint of a segment in the diagram.
• \(\boldsymbol{I}\) is the midpoint of a segment in the diagram.