Question

using the segment addition postulate, which is true?
number line with points a at -6, b at -1, c at 2, d at 8
options:
ab + bc = ad
ab + bc = cd
bc + cd = ad
bc + cd = bd

Explanation:

Step1: Recall Segment Addition Postulate

The segment addition postulate states that if three points \( X \), \( Y \), \( Z \) are collinear and \( Y \) is between \( X \) and \( Z \), then \( XY + YZ = XZ \).

Step2: Identify Coordinates of Points

From the number line:

• \( A \) is at \( -6 \)
• \( B \) is at \( -1 \)
• \( C \) is at \( 2 \)
• \( D \) is at \( 8 \)

Step3: Calculate Lengths of Segments

• \( AB \): Distance between \( A \) and \( B \) is \( | -1 - (-6)| = |5| = 5 \)
• \( BC \): Distance between \( B \) and \( C \) is \( |2 - (-1)| = |3| = 3 \)
• \( CD \): Distance between \( C \) and \( D \) is \( |8 - 2| = |6| = 6 \)
• \( AD \): Distance between \( A \) and \( D \) is \( |8 - (-6)| = |14| = 14 \)
• \( BD \): Distance between \( B \) and \( D \) is \( |8 - (-1)| = |9| = 9 \)

Step4: Check Each Option

• Option 1: \( AB + BC = 5 + 3 = 8 \), \( AD = 14 \). \( 8

eq 14 \), so false.

• Option 2: \( AB + BC = 8 \), \( CD = 6 \). \( 8

eq 6 \), so false.

• Option 3: \( BC + CD = 3 + 6 = 9 \), \( AD = 14 \). \( 9

eq 14 \), so false.

• Option 4: \( BC + CD = 3 + 6 = 9 \), \( BD = 9 \). \( 9 = 9 \), so true.

Answer:

\( \text{BC + CD = BD} \) (the last option, e.g., if options are labeled as D: BC + CD = BD, then D. BC + CD = BD)