Circle P has diameter MT¯¯¯¯¯¯¯¯¯. If MT¯¯¯¯¯¯¯¯¯ intersects chord DR¯¯¯¯¯¯¯¯ at E and m∠TED=90°, then which of the following statements must also be true?(1 point)

Question

Responses

MD¯¯¯¯¯¯¯¯¯≅TD¯¯¯¯¯¯¯¯
Modifying above upper M upper D with bar congruent to Modifying above upper T upper D with bar

m∠MET=90°
m angle upper M upper E upper T equals 90 degrees

EM¯¯¯¯¯¯¯¯¯≅ET¯¯¯¯¯¯¯
Modifying above upper E upper M with bar congruent to Modifying above upper E upper T with bar

ED¯¯¯¯¯¯¯¯≅ER¯¯¯¯¯¯¯¯

Answer 1

To analyze the given problem, we start with some key points derived from the information provided:

1. Circle and Chords: Since MT is the diameter of circle P, any angle inscribed in the circle where the endpoints of the angle lie on the diameter (such as angle \( TED \)) is a right angle. This means that the point \( E \), where \( MT \) intersects chord \( DR \), and creates angle \( TED \), forms a right angle.

2. Properties of Chords and Diameter: In circles, when a diameter intersects a chord, two notable properties emerge:

   • The angles formed by this intersection will create segments that have unique relationships to each other. Specifically, if you have a diameter intersecting a chord, the segment from the center to the intersection point will bisect the chord.

Now, given the angle \( m\angle TED = 90° \), we can draw the following conclusions:

• About the angles: If \( m\angle TED = 90° \), then \( E \) must be the midpoint of the chord \( DR \). In circle geometry, if a diameter bisects a chord at a right angle, it means that the segments from the intersection point (point E) to the endpoints of the chord (points D and R) are equal, which gives us \( ED \cong ER \).

Based on the reasoning above, let's evaluate each option:

1. \( MD \cong TD \): This statement is not necessarily true based on the information provided.

2. \( m\angle MET = 90° \): Since \( MT \) is the diameter and intersects at point \( E \), the angle \( MET \) is not marked as a right angle in this context, hence this statement is not necessarily true.

3. \( EM \cong ET \): This relation can't be established from the given information.

4. \( ED \cong ER \): This is indeed true because when a diameter bisects a chord at a right angle, it implies that the segments created from the intersection point to the endpoints of the chord are equal.

Final Answer:

The statement that must also be true is:

\( ED \cong ER \).