Question

point b is the midpoint of (overline{ac}). which statements about the figure must be true? select three options. (square) (angle dbc) is bisected by ray bd. (square) (angle abc) is bisected by ray bd (square) (bc = \frac{1}{2}ac) (square) (overline{db} cong overline{bc}) (square) (2mangle dbc = mangle abc)

Explanation:

Step 1: Analyze \( \angle ABC \) and ray \( BD \)

Since \( B \) is the midpoint of \( \overline{AC} \) and \( \angle DBC \) is a right angle (from the diagram, \( BD \perp AC \)), \( \angle ABC = 180^\circ \)? Wait, no, \( A - B - C \) is a straight line, so \( \angle ABC = 180^\circ \)? Wait, no, \( BD \) is perpendicular to \( AC \), so \( \angle DBA = \angle DBC = 90^\circ \). So ray \( BD \) splits \( \angle ABC \) (which is \( 180^\circ \)) into two \( 90^\circ \) angles. So \( \angle ABC \) is bisected by \( BD \). So this statement is true.

Step 2: Analyze \( BC=\frac{1}{2}AC \)

Since \( B \) is the midpoint of \( \overline{AC} \), by the definition of midpoint, \( AB = BC \) and \( AC = AB + BC = 2BC \), so \( BC=\frac{1}{2}AC \). This is true.

Step 3: Analyze \( 2m\angle DBC = m\angle ABC \)

We know \( \angle DBC = 90^\circ \) (since \( BD \perp AC \)) and \( \angle ABC = 180^\circ \) (since \( A, B, C \) are colinear). Then \( 2\times90^\circ = 180^\circ \), so \( 2m\angle DBC = m\angle ABC \). This is true.

Step 4: Analyze \( \angle DBC \) bisected by \( BD \)

\( \angle DBC \) is a right angle, and ray \( BD \) is one of its sides, so it can't bisect \( \angle DBC \). This is false.

Step 5: Analyze \( \overline{DB} \cong \overline{BC} \)

We know \( BC \) is a segment on the line \( AC \), and \( DB \) is perpendicular to \( AC \). There's no information to suggest their lengths are equal. So this is false.

Answer:

\( \angle ABC \) is bisected by ray \( BD \), \( BC = \frac{1}{2}AC \), \( 2m\angle DBC = m\angle ABC \)