Question

point e is the midpoint of (overline{ab}) and point f is the midpoint of (overline{cd}). which statements about the figure must be true? select three options. (square) (overline{ab}) is bisected by (overline{cd}). (square) (overline{cd}) is bisected by (overline{ab}). (square) (ae = \frac{1}{2}ab) (square) (ef = \frac{1}{2}ed) (square) (ce + ef = fd)

Explanation:

Step 1: Analyze the midpoint definition

A midpoint of a segment divides it into two equal parts. So, if \( E \) is the midpoint of \( \overline{AB} \), then \( AE = EB=\frac{1}{2}AB \). If \( F \) is the midpoint of \( \overline{CD} \), then \( CF = FD=\frac{1}{2}CD \).

Step 2: Evaluate each statement

• Statement 1: \( \overline{AB} \) is bisected by \( \overline{CD} \)

For a segment to bisect another, it must intersect at the midpoint. \( E \) is the midpoint of \( \overline{AB} \) and \( \overline{CD} \) intersects \( \overline{AB} \) at \( E \), so \( \overline{CD} \) bisects \( \overline{AB} \). This is true.

• Statement 2: \( \overline{CD} \) is bisected by \( \overline{AB} \)

\( \overline{AB} \) intersects \( \overline{CD} \) at \( E \), but \( E \) is not necessarily the midpoint of \( \overline{CD} \) (midpoint of \( \overline{CD} \) is \( F \)). So this is false.

• Statement 3: \( AE=\frac{1}{2}AB \)

By definition of midpoint (\( E \) is midpoint of \( \overline{AB} \)), \( AE = EB=\frac{1}{2}AB \). This is true.

• Statement 4: \( EF=\frac{1}{2}ED \)

There’s no information to support this. \( F \) is midpoint of \( \overline{CD} \), but \( EF \) and \( ED \)’s relationship isn’t guaranteed to be \( EF=\frac{1}{2}ED \). This is false.

• Statement 5: \( CE + EF=FD \)

Since \( F \) is midpoint of \( \overline{CD} \), \( CF = FD \). And \( CF=CE + EF \) (segment addition postulate), so \( CE + EF = FD \). This is true.

Answer:

• \( \overline{AB} \) is bisected by \( \overline{CD} \)
• \( AE = \frac{1}{2}AB \)
• \( CE + EF = FD \)