Question

in circle o, \\( \overline { mp } \\) is a diameter. if \\( m \overgroup { np } \\) is 6 more than 5 times the measure of \\( \overgroup { mn } \\), what is \\( m \overgroup { np }? \\)

Explanation:

Step1: Set up the equation

Let \( m\overarc{MN}=x\). Then \( m\overarc{NP}=5x + 6\). Since \( \overarc{MP}\) is a diameter, \( m\overarc{MN}+m\overarc{NP}=180^{\circ}\). So \( x+(5x + 6)=180\).

Step2: Solve the equation

Simplify the left - hand side: \( 6x+6 = 180\). Subtract 6 from both sides: \( 6x=180 - 6=174\). Divide both sides by 6: \( x=\frac{174}{6}=29\).

Step3: Find \( m\overarc{NP}\)

Substitute \( x = 29\) into \( m\overarc{NP}=5x + 6\). Then \( m\overarc{NP}=5\times29+6=145 + 6=151^{\circ}\).

Answer:

\(151^{\circ}\)