Question

1. if r is the midpoint of \\(\overline{qs}\\), \\(rs = 2x - 4\\), \\(st = 4x - 1\\), and \\(rt = 8x - 43\\), find \\(qs\\).

Explanation:

Step1: Analyze RT composition

RT is RS + ST, so \( RT = RS + ST \).
Substitute values: \( 8x - 43=(2x - 4)+(4x - 1) \)

Step2: Simplify equation

Combine like terms: \( 8x - 43 = 6x - 5 \)
Subtract \( 6x \): \( 2x - 43=-5 \)
Add 43: \( 2x = 38 \)
Divide by 2: \( x = 19 \)

Step3: Find RS length

Substitute \( x = 19 \) into \( RS = 2x - 4 \):
\( RS = 2(19)-4 = 38 - 4 = 34 \)

Step4: Find QS length

Since R is midpoint, \( QS = 2 \times RS \):
\( QS = 2 \times 34 = 68 \)

Answer:

\( 68 \)