Question

look at the diagram and evaluate. which of the statements below is false?
number line with q at 6, r at 15, s at 18, t at 21 (0 to 24 scale)
answer
○ rs > qr ○ qs > rt
○ rs = st ○ qr < qt

Explanation:

First, we identify the positions of each point:

• \( Q = 6 \)
• \( R = 15 \)
• \( S = 18 \)
• \( T = 21 \)

Step 1: Calculate \( RS \), \( QR \)

\( RS = S - R = 18 - 15 = 3 \)
\( QR = R - Q = 15 - 6 = 9 \)
Check \( RS > QR \): \( 3 > 9 \)? No, but wait, maybe I miscalculated. Wait, no: \( RS = 18 - 15 = 3 \), \( QR = 15 - 6 = 9 \). Wait, that would mean \( RS < QR \), but let's check other options too.

Step 2: Calculate \( QS \), \( RT \)

\( QS = S - Q = 18 - 6 = 12 \)
\( RT = T - R = 21 - 15 = 6 \)
Check \( QS > RT \): \( 12 > 6 \)? Yes, that's true.

Step 3: Calculate \( RS \), \( ST \)

\( RS = 18 - 15 = 3 \)
\( ST = T - S = 21 - 18 = 3 \)
Check \( RS = ST \): \( 3 = 3 \)? Yes, that's true.

Step 4: Calculate \( QR \), \( QT \)

\( QR = 15 - 6 = 9 \)
\( QT = T - Q = 21 - 6 = 15 \)
Check \( QR < QT \): \( 9 < 15 \)? Yes, that's true.

Wait, but the first option \( RS > QR \): \( RS = 3 \), \( QR = 9 \), so \( 3 > 9 \) is false. But wait, maybe I mixed up? Wait, no, let's recheck:

Wait, \( Q \) is at 6, \( R \) at 15: \( QR = 15 - 6 = 9 \)

\( R \) at 15, \( S \) at 18: \( RS = 18 - 15 = 3 \)

So \( RS = 3 \), \( QR = 9 \), so \( RS > QR \) is false. But wait, the options: the first option is \( RS > QR \), which is false. But let's check the second option \( QS > RT \):

\( QS = 18 - 6 = 12 \), \( RT = 21 - 15 = 6 \), so \( 12 > 6 \) is true.

\( RS = ST \): \( 3 = 3 \), true.

\( QR < QT \): \( 9 < 15 \), true.

So the false statement is \( RS > QR \). Wait, but the options are:

• \( RS > QR \)
• \( QS > RT \)
• \( RS = ST \)
• \( QR < QT \)

Wait, maybe I made a mistake. Wait, \( Q \) is at 6, \( R \) at 15: \( QR = 15 - 6 = 9 \)

\( R \) at 15, \( S \) at 18: \( RS = 18 - 15 = 3 \)

So \( RS = 3 \), \( QR = 9 \), so \( RS > QR \) is false. So that's the false statement.

Answer:

\( \boldsymbol{RS > QR} \) (the first option)