\( \overline{R S} \) is bisected at \( T \). If \( R S=5 x+66 \) and \( T S=4 x+36 \), what is the length of \( \overline{R T} \) ?

To find the length of \( \overline{R T} \), we start by recognizing that since \( T \) bisects \( \overline{R S} \), it means that \( R T = T S \). This also means that: \[ R S = 2 \times T S \] Given \( R S = 5x + 66 \) and \( T S = 4x + 36 \), we can express the relationship as: \[ 5x + 66 = 2(4x + 36) \] Now, let's expand the right side of the equation: \[ 5x + 66 = 8x + 72 \] Next, we rearrange the equation to isolate \( x \): \[ 5x - 8x = 72 - 66 \] This simplifies to: \[ -3x = 6 \] Dividing both sides by -3 gives: \[ x = -2 \] Now that we have \( x \), we can substitute it back into the expression for \( T S \) to find the length: \[ T S = 4(-2) + 36 = -8 + 36 = 28 \] Since \( R T = T S \): \[ R T = 28 \] Thus, the length of \( \overline{R T} \) is \( \boxed{28} \).