Question

what is $moverset{\frown}{po}$?$\bigcirc$ $128^circLXB0\bigcirc$ $192^circ$$\bigcirc$ $256^circ$the circle has an inscribed arc $overset{\frown}{mo}$ with measure $83^circ$, and an external angle at point $n$ with measure $45^circ$.

Explanation:

Step1: Find arc PMO measure

The inscribed angle over arc MO is $83^\circ$, so arc $m\overset{\frown}{MO} = 2\times83^\circ = 166^\circ$.

Step2: Find arc PO using secant formula

The secant-secant angle formula: $\angle N = \frac{1}{2}(m\overset{\frown}{PO} - m\overset{\frown}{MO})$. Substitute $\angle N=45^\circ$, $m\overset{\frown}{MO}=166^\circ$.
$45^\circ = \frac{1}{2}(m\overset{\frown}{PO} - 166^\circ)$
Multiply both sides by 2: $90^\circ = m\overset{\frown}{PO} - 166^\circ$

Step3: Solve for arc PO

$m\overset{\frown}{PO} = 90^\circ + 166^\circ = 256^\circ$

Answer:

256°