Question

consider the diagram. what is qs? 2 units. 5 units. 17 units. 33 units.

Explanation:

Step1: Identify congruent segments

Since line \( l \) is perpendicular to \( RS \) and \( QS \) and bisects \( RQ \) (marked with equal segments), triangles \( RTS \) and \( QTS \) are congruent, so \( RS = QS \). Thus, \( 3x + 2 = 5x - 8 \).

Step2: Solve for \( x \)

\[

$$\begin{align*} 3x + 2 &= 5x - 8 \\ 2 + 8 &= 5x - 3x \\ 10 &= 2x \\ x &= 5 \end{align*}$$

\]

Step3: Find \( QS \) length

Substitute \( x = 5 \) into \( 5x - 8 \): \( 5(5) - 8 = 25 - 8 = 17 \). Wait, no, \( QS \) is the segment from \( Q \) to \( S \)? Wait, no, the diagram: \( RQ \) is bisected by \( l \) at \( T \), so \( RT = TQ \), and \( l \perp RQ \), so \( RS = QS \) (perpendicular bisector theorem). Wait, but the question is \( QS \)? Wait, no, the options: 2,5,17,33. Wait, maybe I misread. Wait, \( RS = 3x + 2 \), \( QS = 5x - 8 \), and since \( l \) is the perpendicular bisector, \( RS = QS \), so \( 3x + 2 = 5x - 8 \), solve for \( x \): \( 2 + 8 = 5x - 3x \), \( 10 = 2x \), \( x = 5 \). Then \( QS = 5(5) - 8 = 17 \)? Wait, but the options have 17. Wait, but maybe the question is \( QS \) as the length from \( Q \) to \( S \), but maybe I made a mistake. Wait, no, let's check again. Wait, the diagram: \( R \) to \( S \) is \( 3x + 2 \), \( Q \) to \( S \) is \( 5x - 8 \), and \( RT = TQ \), \( l \perp RQ \), so \( RS = QS \), so \( 3x + 2 = 5x - 8 \), \( x = 5 \), so \( QS = 5*5 - 8 = 17 \). So the answer is 17. Wait, but the options: 2,5,17,33. So 17 is an option. Wait, but maybe I misread the question: "What is QS?" Wait, maybe \( QS \) is the segment from \( Q \) to \( S \), which is \( 5x - 8 \), so with \( x = 5 \), that's 17. So the answer is 17.

Wait, but let's check again. Wait, maybe the question is \( QS \) as in the length from \( Q \) to \( S \), which is \( 5x - 8 \), and we found \( x = 5 \), so \( 5*5 - 8 = 17 \). So the answer is 17.

Answer:

17 units.