Question

if tu = 6 units, what must be true?

su + ut = rt
rt + tu = rs
rs + su = ru
tu + us = rs

Explanation:

Identify collinear points and segment lengths

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Apply Segment Addition Postulate

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Evaluate the given options

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Let's check the options:

1. \(SU + UT = RT\): False, \(SU + UT = ST\).
2. \(RT + TU = RS\): False, \(RT + TU = RU

eq RS\).

1. \(RS + SU = RU\): False, \(RU + US = RS\).
2. \(TU + US = RS\): Let's check. Since \(T, U, S\) are collinear in that order, \(TU + US = TS\). Since \(TS = 12\) and \(TU = 6\), we have \(US = TS - TU = 12 - 6 = 6\). Thus, \(TU + US = 6 + 6 = 12\), which is \(TS\), not \(RS = 24\).

Wait, let's re-evaluate the Segment Addition Postulate for the options:

• Option 1: \(SU + UT = RT\). Since \(U\) is between \(T\) and \(S\), \(SU + UT = ST\). Since \(ST = 12\) and \(RT = 12\), \(SU + UT = RT\) is true!

Let's verify:
\(ST = RS - RT = 24 - 12 = 12\).
Since \(U\) is on the segment \(TS\), \(UT + US = ST = 12\).
Since \(RT = 12\), we have \(UT + US = RT\), which is \(SU + UT = RT\).
This must be true.

Answer:

• (A) SU + UT = RT (Correct answer)
• (B) RT + TU = RS
• (C) RS + SU = RU
• (D) TU + US = RS