Question

what is the length of segment sr? \boxed{\\_} units (diagram: points r, t, q on a horizontal line with rt = tq (marked by ticks), line l perpendicular to rq at t, point s on line l below t. segments sr = 2x + 8, sq = 8x - 4)

Explanation:

Step1: Identify congruent segments

Since line \( l \) is the perpendicular bisector of \( RQ \) (as \( RT = TQ \) and \( \angle STR = 90^\circ \)), triangles \( SRT \) and \( SQT \) are congruent by SAS. Thus, \( SR = SQ \).

Step2: Set up equation

Set \( 2x + 8 = 8x - 4 \) (because \( SR = SQ \)).

Step3: Solve for \( x \)

Subtract \( 2x \) from both sides: \( 8 = 6x - 4 \).
Add 4 to both sides: \( 12 = 6x \).
Divide by 6: \( x = 2 \).

Step4: Calculate \( SR \)

Substitute \( x = 2 \) into \( 2x + 8 \): \( 2(2) + 8 = 4 + 8 = 12 \).

Answer:

12