Question

point b is the midpoint of \\( \overline { a c } \\).
which statements about the figure must be true? select
three options.
\\( \square \angle d b c \\) is bisected by ray bd.
\\( \square \angle a b c \\) is bisected by ray bd.
\\( \square b c = \frac { 1 } { 2 } a c \\)
\\( \square \overline { d b } \cong \overline { b c } \\)
\\( \square 2 m \angle d b c = m \angle a b c \\)

Explanation:

Step1: Recall the definition of a mid - point

If \(B\) is the mid - point of \(\overline{AC}\), then \(AB = BC\) and \(BC=\frac{1}{2}AC\) (by the mid - point formula \(AB + BC=AC\) and \(AB = BC\)).

Step2: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal parts. Since \(B\) is a point and \(BD\) is a ray such that \(\angle ABD=\angle DBC = 90^{\circ}\), ray \(BD\) bisects \(\angle ABC\) (because \(\angle ABC=\angle ABD+\angle DBC\) and \(\angle ABD=\angle DBC\)).

Step3: Analyze each option

• For \(\angle DBC\) is bisected by ray \(BD\): A ray cannot bisect itself. So this is false.
• For \(\angle ABC\) is bisected by ray \(BD\): \(\angle ABC = 180^{\circ}\) (a straight - angle) and \(\angle ABD=\angle DBC=90^{\circ}\). So this is true.
• For \(BC=\frac{1}{2}AC\): By the mid - point property (\(AB = BC\) and \(AB+BC = AC\)), \(BC=\frac{1}{2}AC\). So this is true.
• For \(\overline{DB}\cong\overline{BC}\): There is no information about the length of \(DB\) and \(BC\) (one is a vertical segment and the other is a horizontal segment from the mid - point). So this is false.
• For \(2m\angle DBC=m\angle ABC\): Since \(m\angle DBC = 90^{\circ}\) and \(m\angle ABC=180^{\circ}\), \(2\times90^{\circ}=180^{\circ}\). But \(\angle ABC\) is bisected by \(BD\), the correct relationship is based on the bisector of \(\angle ABC\) (not \(2m\angle DBC=m\angle ABC\) in the sense of bisecting \(\angle DBC\)). The key correct facts come from mid - point and angle - bisector definitions.

Answer:

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