Question

which statement is true about the given information?
line segment with points b, c, d, e and tick marks between b-c, c-d, d-e
options:

• ( \text{bd} = \frac{1}{2}\text{bc} )
• ( \text{bc} = \frac{1}{2}\text{be} )
• ( overline{\text{bd}} cong overline{\text{ce}} )
• ( overline{\text{bc}} cong overline{\text{bd}} )

Explanation:

Step1: Analyze the segment markings

The diagram shows that \( BC = CD = DE \) (since the tick marks indicate equal segments). Let's denote the length of each equal segment (e.g., \( BC \)) as \( x \). Then \( BD = BC + CD = x + x = 2x \), and \( CE = CD + DE = x + x = 2x \). Also, \( BE = BC + CD + DE = x + x + x = 3x \).

Step2: Evaluate each option

• Option 1: \( BD=\frac{1}{2}BC \). But \( BD = 2x \) and \( BC = x \), so \( BD = 2BC \), not \( \frac{1}{2}BC \). Eliminate.
• Option 2: \( BC=\frac{1}{2}BE \). \( BE = 3x \), so \( \frac{1}{2}BE=\frac{3x}{2} \), and \( BC = x \). \( x

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

• Option 3: \( \overline{BD}\cong\overline{CE} \). \( BD = 2x \) and \( CE = 2x \), so their lengths are equal, meaning the segments are congruent. This is true.
• Option 4: \( \overline{BC}\cong\overline{BD} \). \( BC = x \) and \( BD = 2x \), so their lengths are not equal. Eliminate.

Answer:

\( \overline{BD} \cong \overline{CE} \) (the third option)