Question

congruence stations
if line ( n ) bisects ( overline{qr} ), find ( qp ).
diagram: points ( q ), ( p ), ( r ) on a straight line; ( qp = 3x + 5 ), ( pr = 5x - 19 ); line ( n ) intersects at ( p ).

Explanation:

Step1: Set QP equal to PR (bisect definition)

Since line \( n \) bisects \( \overline{QR} \), \( QP = PR \). So we set the expressions equal: \( 3x + 5 = 5x - 19 \).

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 5 = 2x - 19 \).
Add 19 to both sides: \( 24 = 2x \).
Divide by 2: \( x = 12 \).

Step3: Find QP by substituting \( x \)

Substitute \( x = 12 \) into \( QP = 3x + 5 \):
\( QP = 3(12) + 5 = 36 + 5 = 41 \).

Answer:

\( 41 \)