Question

Circle $P$ has diameter $\overline{MT}$. If $\overline{MT}$ intersects chord $\overline{DR}$ at $E$ and $m\angle TED = 90^\circ$, then which of the following statements must also be true?Responses$\overline{ED} \cong \overline{ER}$ $m\angle MET = 90^\circ$$\overline{MD} \cong \overline{TD}$$\overline{EM} \cong \overline{ET}$

Answer 1

Answer: ED ≅ ER.

Reason: MT is a diameter so its line passes through the circle's center. ET (part of MT) is perpendicular to chord DR at E, so the line through the center is perpendicular to the chord. A radius/line through the center that is perpendicular to a chord bisects that chord, so E is the midpoint of DR and ED = ER.

The others need not be true: m∠MET is a straight angle (180°), not 90°; MD and TD need not be equal; EM and ET are equal only if E is the center, which is not given.