Question
ED¯¯¯¯¯¯¯¯
is a perpendicular bisector of GF¯¯¯¯¯¯¯¯
, and they intersect at point P. Which of the following statements must be true? Select the TWO answers that are correct.(1 point)
Responses
∠FDP≅∠EPG
∠FDP≅∠EPG
∠DPF≅∠DPG
∠DPF≅∠DPG
GP¯¯¯¯¯¯¯¯≅FP¯¯¯¯¯¯¯¯
line segment cap g cap p is congruent to line segment cap f cap p
DP¯¯¯¯¯¯¯¯≅EP¯¯¯¯¯¯¯¯
line segment cap d cap p is congruent to line segment cap e cap p
GF¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯
line segment cap g cap f is congruent to line segment cap e cap d
is a perpendicular bisector of GF¯¯¯¯¯¯¯¯
, and they intersect at point P. Which of the following statements must be true? Select the TWO answers that are correct.(1 point)
Responses
∠FDP≅∠EPG
∠FDP≅∠EPG
∠DPF≅∠DPG
∠DPF≅∠DPG
GP¯¯¯¯¯¯¯¯≅FP¯¯¯¯¯¯¯¯
line segment cap g cap p is congruent to line segment cap f cap p
DP¯¯¯¯¯¯¯¯≅EP¯¯¯¯¯¯¯¯
line segment cap d cap p is congruent to line segment cap e cap p
GF¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯
line segment cap g cap f is congruent to line segment cap e cap d
Answers
Answered by
GPT-4o mini
To analyze the situation given, we know that \( ED \) is a perpendicular bisector of \( GF \) and they intersect at point \( P \). This means that:
1. \( P \) is the midpoint of \( GF \).
2. \( ED \) intersects \( GF \) at a right angle.
With that in mind, let’s evaluate the possible statements:
1. **\( \angle FDP \cong \angle EPG \)**: This statement suggests that the angles formed by segment \( DF \) and segment \( EP \) are congruent, which is true because both angles are formed at point \( P \) where the perpendicular bisector intersects \( GF \).
2. **\( \angle DPF \cong \angle DPG \)**: This statement appears to be incorrect because they are not angles formed by segments that share a common vertex and line but rather could correspond to two different segments where the relationship is not clear.
3. **\( GP \cong FP \)**: This statement must be true because \( P \) is the midpoint of \( GF \). By the definition of a midpoint, it divides the segment into two equal parts.
4. **\( DP \cong EP \)**: This statement is not necessarily true based solely on the given information. We cannot conclude anything about the lengths of \( DP \) and \( EP \) unless more information about points \( D \) and \( E \) is provided.
5. **\( GF \cong ED \)**: This statement is also not necessarily true since we cannot infer anything about the lengths of \( GF \) and \( ED \) just from knowing one is a bisector of the other.
### The Correct Statements:
The two statements that must be true are:
- \( \angle FDP \cong \angle EPG \)
- \( GP \cong FP \)
Therefore, the two correct answers are:
1. \( \angle FDP \cong \angle EPG \)
2. \( GP \cong FP \)
1. \( P \) is the midpoint of \( GF \).
2. \( ED \) intersects \( GF \) at a right angle.
With that in mind, let’s evaluate the possible statements:
1. **\( \angle FDP \cong \angle EPG \)**: This statement suggests that the angles formed by segment \( DF \) and segment \( EP \) are congruent, which is true because both angles are formed at point \( P \) where the perpendicular bisector intersects \( GF \).
2. **\( \angle DPF \cong \angle DPG \)**: This statement appears to be incorrect because they are not angles formed by segments that share a common vertex and line but rather could correspond to two different segments where the relationship is not clear.
3. **\( GP \cong FP \)**: This statement must be true because \( P \) is the midpoint of \( GF \). By the definition of a midpoint, it divides the segment into two equal parts.
4. **\( DP \cong EP \)**: This statement is not necessarily true based solely on the given information. We cannot conclude anything about the lengths of \( DP \) and \( EP \) unless more information about points \( D \) and \( E \) is provided.
5. **\( GF \cong ED \)**: This statement is also not necessarily true since we cannot infer anything about the lengths of \( GF \) and \( ED \) just from knowing one is a bisector of the other.
### The Correct Statements:
The two statements that must be true are:
- \( \angle FDP \cong \angle EPG \)
- \( GP \cong FP \)
Therefore, the two correct answers are:
1. \( \angle FDP \cong \angle EPG \)
2. \( GP \cong FP \)