Question
in the diagram below, (DE with a line above with an arrow on both sides) is the perpendicular bisector of (FG with a line above with no arrows). which of the following statements must be true? (the diagram below shows a line segment FG and a line DE perpendicular to it with a shared vertex of H). here are the following possible answers, Check all that apply. a. ∠ DHF is a right angle, b. (DE with a line with arrows on both ends) is perpendicular to FD with a line above it with no arrows, C. H is the midpoint to FG with a line above it with no arrows, d. (DH with a line above it with no arrows) ≅ (EH with a line above it with no arrows), e. (FH with a line above it with no arrows) ≅ (GH with a line above it with no arrows), F. (FG with a line above it with no arrows) bisects )DE with a line with arrows on both ends)
Answers
Answer
but the question is DE is a perpendicular bisector of FG
Answered by
GPT-4o mini
Based on the description of the diagram and the properties of geometric figures, we can analyze each statement to determine which must be true given that line \( DE \) is the perpendicular bisector of segment \( FG \).
1. **Statement a: \( \angle DHF \) is a right angle.**
- This statement is true because a perpendicular bisector creates right angles with the segment it bisects.
2. **Statement b: \( DE \) is perpendicular to \( FD \).**
- This statement is not necessarily true. While \( DE \) is perpendicular to \( FG \), we do not have enough information to conclude that it is also perpendicular to \( FD \).
3. **Statement c: \( H \) is the midpoint of \( FG \).**
- This statement is true because the definition of a perpendicular bisector states that it divides the segment it bisects into two equal parts at the midpoint.
4. **Statement d: \( DH \cong EH \).**
- This statement is true. Since \( DE \) is the perpendicular bisector of \( FG \), the segments \( DH \) and \( EH \) are equal as they are both segments from point \( D \) to point \( H \) and from point \( E \) to point \( H \), respectively.
5. **Statement e: \( FH \cong GH \).**
- This statement is true as well. Because \( H \) is the midpoint of \( FG \) (as established in part c), the segments \( FH \) and \( GH \) are equal.
6. **Statement f: \( FG \) bisects \( DE \).**
- This statement is not true. The segment \( FG \) does not bisect line \( DE \); rather, \( DE \) bisects \( FG \).
**Summary of Statements that Must Be True:**
- **a:** True
- **c:** True
- **d:** True
- **e:** True
Thus, the statements that must be true are: **a, c, d, e**.
1. **Statement a: \( \angle DHF \) is a right angle.**
- This statement is true because a perpendicular bisector creates right angles with the segment it bisects.
2. **Statement b: \( DE \) is perpendicular to \( FD \).**
- This statement is not necessarily true. While \( DE \) is perpendicular to \( FG \), we do not have enough information to conclude that it is also perpendicular to \( FD \).
3. **Statement c: \( H \) is the midpoint of \( FG \).**
- This statement is true because the definition of a perpendicular bisector states that it divides the segment it bisects into two equal parts at the midpoint.
4. **Statement d: \( DH \cong EH \).**
- This statement is true. Since \( DE \) is the perpendicular bisector of \( FG \), the segments \( DH \) and \( EH \) are equal as they are both segments from point \( D \) to point \( H \) and from point \( E \) to point \( H \), respectively.
5. **Statement e: \( FH \cong GH \).**
- This statement is true as well. Because \( H \) is the midpoint of \( FG \) (as established in part c), the segments \( FH \) and \( GH \) are equal.
6. **Statement f: \( FG \) bisects \( DE \).**
- This statement is not true. The segment \( FG \) does not bisect line \( DE \); rather, \( DE \) bisects \( FG \).
**Summary of Statements that Must Be True:**
- **a:** True
- **c:** True
- **d:** True
- **e:** True
Thus, the statements that must be true are: **a, c, d, e**.
Answered by
GPT-4o mini
Thank you for the clarification. Since \( DE \) is explicitly stated as the perpendicular bisector of segment \( FG \), we can analyze the statements again with that definition in mind.
### Given:
- \( DE \) is the **perpendicular bisector** of \( FG \).
- This means:
- \( DE \) intersects \( FG \) at its midpoint \( H \).
- \( \angle DHF \) and \( \angle EHG \) are right angles (90 degrees).
### Analysis of Each Statement:
1. **Statement a: \( \angle DHF \) is a right angle.**
- **True.** Since \( DE \) is the perpendicular bisector of \( FG \), it forms right angles with \( FG \) at point \( H \).
2. **Statement b: \( DE \) is perpendicular to \( FD \).**
- **Not necessarily true.** \( DE \) is perpendicular to \( FG \) but not to \( FD \) without additional information indicating that \( H \) is on \( FD \).
3. **Statement c: \( H \) is the midpoint of \( FG \).**
- **True.** By definition of a perpendicular bisector, \( H \) is the midpoint of segment \( FG \).
4. **Statement d: \( DH \cong EH \).**
- **True.** Since \( DE \) bisects segment \( FG \) and \( H \) is the midpoint, the segments \( DH \) and \( EH \) are equal.
5. **Statement e: \( FH \cong GH \).**
- **True.** By definition of a midpoint, the segments \( FH \) and \( GH \) are equal.
6. **Statement f: \( FG \) bisects \( DE \).**
- **Not true.** A segment cannot bisect a line; rather, a line can bisect a segment. Therefore, this statement is not valid.
### Summary of True Statements:
Given that \( DE \) is the perpendicular bisector of \( FG \):
- **a:** True
- **c:** True
- **d:** True
- **e:** True
The correct statements that must be true are: **a, c, d, and e.** Thank you for your patience as we clarified this!
### Given:
- \( DE \) is the **perpendicular bisector** of \( FG \).
- This means:
- \( DE \) intersects \( FG \) at its midpoint \( H \).
- \( \angle DHF \) and \( \angle EHG \) are right angles (90 degrees).
### Analysis of Each Statement:
1. **Statement a: \( \angle DHF \) is a right angle.**
- **True.** Since \( DE \) is the perpendicular bisector of \( FG \), it forms right angles with \( FG \) at point \( H \).
2. **Statement b: \( DE \) is perpendicular to \( FD \).**
- **Not necessarily true.** \( DE \) is perpendicular to \( FG \) but not to \( FD \) without additional information indicating that \( H \) is on \( FD \).
3. **Statement c: \( H \) is the midpoint of \( FG \).**
- **True.** By definition of a perpendicular bisector, \( H \) is the midpoint of segment \( FG \).
4. **Statement d: \( DH \cong EH \).**
- **True.** Since \( DE \) bisects segment \( FG \) and \( H \) is the midpoint, the segments \( DH \) and \( EH \) are equal.
5. **Statement e: \( FH \cong GH \).**
- **True.** By definition of a midpoint, the segments \( FH \) and \( GH \) are equal.
6. **Statement f: \( FG \) bisects \( DE \).**
- **Not true.** A segment cannot bisect a line; rather, a line can bisect a segment. Therefore, this statement is not valid.
### Summary of True Statements:
Given that \( DE \) is the perpendicular bisector of \( FG \):
- **a:** True
- **c:** True
- **d:** True
- **e:** True
The correct statements that must be true are: **a, c, d, and e.** Thank you for your patience as we clarified this!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.