Question

s is the midpoint of \\(\overline{rt}\\). if \\(st = x + 4\\) and \\(rt = 6x + 4\\), what is \\(rt\\)?
diagram: segment with endpoints r and t, midpoint s. st is labeled \\(x + 4\\), rt is labeled \\(6x + 4\\)
simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( S \) is the midpoint of \( \overline{RT} \), \( RT = 2 \times ST \). So we have the equation \( 6x + 4 = 2(x + 4) \).

Step2: Solve for \( x \)

Expand the right side: \( 6x + 4 = 2x + 8 \).
Subtract \( 2x \) from both sides: \( 6x - 2x + 4 = 8 \), which simplifies to \( 4x + 4 = 8 \).
Subtract 4 from both sides: \( 4x = 8 - 4 = 4 \).
Divide both sides by 4: \( x = \frac{4}{4} = 1 \).

Step3: Find \( RT \)

Substitute \( x = 1 \) into \( RT = 6x + 4 \). So \( RT = 6(1) + 4 = 6 + 4 = 10 \).

Answer:

\( 10 \)