Question

in circle o, mp is a diameter.
if m\overarc{np} is 6 more than 5 times the measure of \overarc{mn}, wh
is m\overarc{np}?
\\( \circ 139 ^ { \circ } \\)
\\( \circ 145 ^ { \circ } \\)
\\( \circ 151 ^ { \circ } \\)
\\( \circ 174 ^ { \circ } \\)

Explanation:

Step1: Set up the equation

Let \( m\widehat{MN}=x \). Then \( m\widehat{NP}=5x + 6 \). Since \( \widehat{MN}\) and \( \widehat{NP}\) are arcs that make up a semicircle (because \( MP\) is a diameter), \( m\widehat{MN}+m\widehat{NP}=180^{\circ}\). 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 \( m\widehat{NP}\)

Substitute \( x = 29\) into the expression for \( m\widehat{NP}\). \( m\widehat{NP}=5x+6\). So \( m\widehat{NP}=5\times29 + 6=145+6=151^{\circ}\).

Answer:

\(151^{\circ}\) (the third option)