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)

Question

GPT-4o mini · Answer

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 $.

Statement a: $ \angle DHF $ is a right angle.
- This statement is true because a perpendicular bisector creates right angles with the segment it bisects.
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 $.
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.
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.
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.
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.

Answer

but the question is DE is a perpendicular bisector of FG

GPT-4o mini · Answer

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!