Question
Select the choice that indicates AB←→%0D%0A𝐴%0D%0A𝐵%0D%0A↔%0D%0A is the perpendicular bisector of CD←→%0D%0A𝐶%0D%0A𝐷%0D%0A↔%0D%0A if they intersect at P%0D%0A𝑃%0D%0A.(1 point)%0D%0AResponses%0D%0A%0D%0AAP¯¯¯¯¯¯¯¯≅BP¯¯¯¯¯¯¯¯%0D%0A𝐴%0D%0A𝑃%0D%0A¯%0D%0A≅%0D%0A𝐵%0D%0A𝑃%0D%0A¯%0D%0AModifying above upper A upper P with bar congruent to Modifying above upper B upper P with bar%0D%0A%0D%0ACD¯¯¯¯¯¯¯¯⊥AB¯¯¯¯¯¯¯¯%0D%0A𝐶%0D%0A𝐷%0D%0A¯%0D%0A⊥%0D%0A𝐴%0D%0A𝐵%0D%0A¯%0D%0AModifying above upper C upper D with bar perpendicular to Modifying above upper A upper B with bar%0D%0A%0D%0AAC¯¯¯¯¯¯¯¯⊥BD¯¯¯¯¯¯¯¯%0D%0A𝐴%0D%0A𝐶%0D%0A¯%0D%0A⊥%0D%0A𝐵%0D%0A𝐷%0D%0A¯%0D%0A and AP¯¯¯¯¯¯¯¯≅BP¯¯¯¯¯¯¯¯%0D%0A𝐴%0D%0A𝑃%0D%0A¯%0D%0A≅%0D%0A𝐵%0D%0A𝑃%0D%0A¯%0D%0AModifying above upper A upper C with bar perpendicular to Modifying above upper B upper D with bar and Modifying above upper A upper P with bar congruent to Modifying above upper B upper P with bar%0D%0A%0D%0AAB¯¯¯¯¯¯¯¯⊥CD¯¯¯¯¯¯¯¯%0D%0A𝐴%0D%0A𝐵%0D%0A¯%0D%0A⊥%0D%0A𝐶%0D%0A𝐷%0D%0A¯%0D%0A and CP¯¯¯¯¯¯¯¯≅DP¯¯¯¯¯¯¯¯
Answers
GPT-4o mini
To determine which choice correctly indicates that line segment AB is the perpendicular bisector of line segment CD, we must analyze the properties of a perpendicular bisector. By definition, a segment is a perpendicular bisector of another segment if it meets the following criteria:
1. It intersects the segment at its midpoint.
2. It is perpendicular to that segment.
Given the point of intersection \( P \), we will evaluate the provided options:
1. **\( \overline{AP} \cong \overline{BP} \)**: This indicates that the segments from points A and B to point P (where they intersect) are congruent. This suggests that P is the midpoint of segment AB, which is necessary for AB to be the perpendicular bisector of CD.
2. **\( \overline{CD} \perp \overline{AB} \)**: This conveys that line segment CD is perpendicular to line segment AB. This correctly states the perpendicularity aspect but is incorrectly phrased since it refers to CD being perpendicular to AB instead of AB being perpendicular to CD.
3. **\( \overline{AC} \perp \overline{BD} \)**: This indicates that segment AC is perpendicular to segment BD, but does not contribute to establishing AB as a perpendicular bisector of CD.
4. **\( \overline{AB} \perp \overline{CD} \) and \( \overline{CP} \cong \overline{DP} \)**: This option asserts both that AB is perpendicular to CD and that P is the midpoint of CD, fulfilling both conditions needed for AB to be the perpendicular bisector of CD.
After evaluating these options, the response indicating that AB is the perpendicular bisector of CD is:
**\( \overline{AB} \perp \overline{CD} \) and \( \overline{CP} \cong \overline{DP} \)**.
1. It intersects the segment at its midpoint.
2. It is perpendicular to that segment.
Given the point of intersection \( P \), we will evaluate the provided options:
1. **\( \overline{AP} \cong \overline{BP} \)**: This indicates that the segments from points A and B to point P (where they intersect) are congruent. This suggests that P is the midpoint of segment AB, which is necessary for AB to be the perpendicular bisector of CD.
2. **\( \overline{CD} \perp \overline{AB} \)**: This conveys that line segment CD is perpendicular to line segment AB. This correctly states the perpendicularity aspect but is incorrectly phrased since it refers to CD being perpendicular to AB instead of AB being perpendicular to CD.
3. **\( \overline{AC} \perp \overline{BD} \)**: This indicates that segment AC is perpendicular to segment BD, but does not contribute to establishing AB as a perpendicular bisector of CD.
4. **\( \overline{AB} \perp \overline{CD} \) and \( \overline{CP} \cong \overline{DP} \)**: This option asserts both that AB is perpendicular to CD and that P is the midpoint of CD, fulfilling both conditions needed for AB to be the perpendicular bisector of CD.
After evaluating these options, the response indicating that AB is the perpendicular bisector of CD is:
**\( \overline{AB} \perp \overline{CD} \) and \( \overline{CP} \cong \overline{DP} \)**.