Question

1. if r is the midpoint of \\(\overline{qs}\\), find rs given qs = 6x - 2. find rs.

there is a line segment with points q, r, s. the length from q to r is 2x + 10

Explanation:

Step1: Use midpoint definition

Since \( R \) is the midpoint of \( \overline{QS} \), \( QR = RS \) and \( QS=QR + RS = 2QR \). We know \( QR = 2x + 10 \) and \( QS=6x - 2 \). So \( 6x-2=2(2x + 10) \)

Step2: Solve for \( x \)

Expand the right side: \( 6x-2 = 4x+20 \)
Subtract \( 4x \) from both sides: \( 6x - 4x-2=4x - 4x+20 \) → \( 2x-2 = 20 \)
Add 2 to both sides: \( 2x-2 + 2=20 + 2 \) → \( 2x=22 \)
Divide by 2: \( x = 11 \)

Step3: Find \( RS \)

Since \( RS = QR=2x + 10 \), substitute \( x = 11 \): \( RS=2(11)+10=22 + 10=32 \)

Answer:

\( 32 \)