Question

if tu = 6 units, what must be true?
options:
su + ut = rt
rt + tu = rs
rs + su = ru
tu + us = rs
(diagram: points r, t, u, s on a line segment; rt labeled 12 units, rs labeled 24 units)

Explanation:

To solve this, we analyze the line segment \( RS \) with points \( T \) and \( U \) on it. We know \( RS = 24 \) units, \( RT = 12 \) units (so \( T \) is the midpoint of \( RS \)), and \( TU = 6 \) units. Let's check each option:

Option 1: \( SU + UT = RT \)

• \( SU \): Since \( RT = 12 \), \( TU = 6 \), then \( US = RS - RT - TU = 24 - 12 - 6 = 6 \) units. So \( SU = 6 \), \( UT = 6 \). Then \( SU + UT = 6 + 6 = 12 \), and \( RT = 12 \). Wait, but let's check other options too.

Option 2: \( RT + TU = RS \)

• \( RT = 12 \), \( TU = 6 \), so \( RT + TU = 18 \), but \( RS = 24 \). This is false.

Option 3: \( RS + SU = RU \)

• \( RS = 24 \), \( SU = 6 \), so \( RS + SU = 30 \), but \( RU = RT + TU = 12 + 6 = 18 \). False.

Option 4: \( TU + US = RS \)

• \( TU = 6 \), \( US = 6 \), so \( TU + US = 12 \), but \( RS = 24 \). False. Wait, earlier for Option 1, \( SU = 6 \), \( UT = 6 \), so \( SU + UT = 12 \), and \( RT = 12 \). But wait, let's re - examine the points. The order of the points is \( R - T - U - S \) (since \( RT = 12 \), \( RS = 24 \), so \( T \) is the midpoint, \( TU = 6 \), so \( U \) is between \( T \) and \( S \), and \( US = 6 \)). So \( SU \) is from \( S \) to \( U \), which is 6, \( UT \) is from \( U \) to \( T \), which is 6. Then \( SU+UT = 6 + 6=12\), and \( RT = 12 \). But wait, maybe I made a mistake in the order. Wait, the line is \( R---T---U---S \), with \( RT = 12 \), \( RS = 24 \), so \( TS=12 \). \( TU = 6 \), so \( US = TS - TU=12 - 6 = 6 \). Now, let's check the options again:

• Option 1: \( SU+UT \): \( SU = 6 \) (from \( S \) to \( U \)), \( UT = 6 \) (from \( U \) to \( T \)), so \( SU + UT=12 \), and \( RT = 12 \). So \( SU + UT=RT \) is true? Wait, but let's check the other options again. Wait, maybe the correct option is the first one. But wait, let's re - evaluate:

Wait, the problem says "what must be true". Let's check the segment addition postulate. The segment addition postulate states that if \( A - B - C \) are collinear, then \( AB+BC = AC \).

For the points \( R - T - U - S \):

• \( RT+TU + US=RS \) (since \( R \) to \( T \) to \( U \) to \( S \) is the whole segment \( RS \)).

Now let's check each option:

1. \( SU + UT=RT \): \( SU = 6 \), \( UT = 6 \), so \( SU + UT = 12 \), and \( RT = 12 \). So this is true? Wait, but \( SU+UT = ST \) (since \( S - U - T \), so \( SU + UT=ST \)). And \( ST = RS - RT=24 - 12 = 12 \), and \( RT = 12 \), so \( SU + UT=RT \) is true.

1. \( RT + TU=RS \): \( RT = 12 \), \( TU = 6 \), \( RT + TU = 18

eq24 = RS \). False.

1. \( RS + SU=RU \): \( RS = 24 \), \( SU = 6 \), \( RS+SU = 30 \), \( RU=RT + TU = 12 + 6 = 18

eq30 \). False.

1. \( TU + US=RS \): \( TU = 6 \), \( US = 6 \), \( TU + US = 12

eq24 = RS \). False.

So the correct option is the first one: \( \boldsymbol{SU + UT = RT} \)

Answer:

A. \( SU + UT = RT \)