Question

if tu = 6 units, what must be true? rs + su = ru tu + us = rs su + ut = rt rt + tu = rs

Explanation:

Step1: Recall segment - addition postulate

The segment - addition postulate states that if we have three collinear points \(A\), \(B\), and \(C\) where \(B\) is between \(A\) and \(C\), then \(AB + BC=AC\).
In the given line - segment \(RS\) with points \(R\), \(T\), \(U\), and \(S\) in order, \(RT + TU=RS\) because \(T\) is between \(R\) and \(S\).
We know that \(RT = 12\) units and \(TU = 6\) units, and \(RS=24\) units. Also, \(RS\) is composed of \(RT\) and \(TU\) and \(US\).

Step2: Analyze each option

• Option 1: \(RS + SU=RU\) is incorrect. By the segment - addition postulate, \(RU+SU = RS\).
• Option 2: \(TU + US=RS\) is incorrect. Since \(RT+TU + US=RS\) and \(RT = 12\), \(TU = 6\), \(US=RS-(RT + TU)=24-(12 + 6)=6\), and \(TU+US=6 + 6=12

eq RS\).

• Option 3: \(SU + UT=RT\) is incorrect. \(SU=6\), \(UT = 6\), \(RT = 12\), but the correct relationship based on the order of points is \(RT+TU+US=RS\).
• Option 4: \(RT + TU=RS\). Given \(RT = 12\) units and \(TU = 6\) units, and \(RS=12 + 6=18\) (but we also know from the figure that \(RS = 24\) units and \(RT+TU\) represents the sum of two sub - segments of \(RS\)). According to the segment - addition postulate for collinear points \(R\), \(T\), and \(S\) where \(T\) is between \(R\) and \(S\), \(RT + TU=RS\).

Answer:

\(RT + TU=RS\)