Question

consider the diagram. what is qs? 2 units. 5 units 17 units. 33 units. diagram: line l perpendicular to segment rq at t, rt and tq marked as equal (congruent segments), point s on line l, rs = 3x + 2, qs = 5x − 8

Explanation:

Step1: Identify congruent triangles

Since \( l \) is perpendicular to \( RQ \) and \( T \) is the midpoint of \( RQ \) (marked by equal segments), triangles \( RTS \) and \( QTS \) are congruent (by SAS: right angle, \( RT = QT \), common side \( ST \)). Thus, \( RS = QS \).

Step2: Set up equation for \( x \)

Set \( 3x + 2 = 5x - 8 \) (since \( RS = QS \)).

Step3: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 2 = 2x - 8 \).
Add 8 to both sides: \( 10 = 2x \).
Divide by 2: \( x = 5 \).

Step4: Calculate \( QS \)

Substitute \( x = 5 \) into \( 5x - 8 \): \( 5(5) - 8 = 25 - 8 = 17 \).

Answer:

17 units.