Question

which statement is true about the given information?
bd = \frac{1}{2}bc
bc = \frac{1}{2}be
\overline{bd} \cong \overline{ce}
\overline{bc} \cong \overline{bd}

Explanation:

Step1: Analyze the line segments

From the diagram, we can see that \(BC = CD=DE\). Let \(BC = x\), then \(CD=x\), \(DE = x\).

Step2: Calculate lengths

• \(BD=BC + CD=x + x=2x\).
• \(BE=BC + CD+DE=x + x+x = 3x\).
• \(\overline{BD}=BC + CD\), \(\overline{CE}=CD + DE\). Since \(BC = CD=DE\), we have \(BD=CE\) (by substitution \(BD=x + x\) and \(CE=x + x\)), so \(\overline{BD}\cong\overline{CE}\) (by the definition of congruent segments, segments with equal lengths are congruent).

• For \(BD=\frac{1}{2}BC\), since \(BD = 2x\) and \(BC=x\), \(2x

eq\frac{1}{2}x\).

• For \(BC=\frac{1}{2}BE\), since \(BC=x\) and \(BE = 3x\), \(x

eq\frac{3x}{2}\).

• For \(\overline{BC}\cong\overline{BD}\), since \(BC=x\) and \(BD = 2x\), \(x

eq2x\).

Answer:

\(\overline{BD}\cong\overline{CE}\)