Question

what is ( moverarc{po} )? ( 128^circ ) ( 173^circ ) ( 192^circ ) ( 256^circ )

Explanation:

Step1: Recall the secant - tangent angle formula

The measure of an angle formed by a secant and a tangent (or two secants) outside a circle is equal to half the difference of the measures of the intercepted arcs. The formula is \(m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\). We know that \(m\angle N = 45^{\circ}\) and \(m\widehat{MO}=83^{\circ}\), and we need to find \(m\widehat{PO}\).

Step2: Substitute the known values into the formula

Substitute \(m\angle N = 45^{\circ}\) and \(m\widehat{MO}=83^{\circ}\) into the formula \(45^{\circ}=\frac{1}{2}(m\widehat{PO}-83^{\circ})\).

Step3: Solve for \(m\widehat{PO}\)

First, multiply both sides of the equation by 2: \(2\times45^{\circ}=m\widehat{PO}-83^{\circ}\), which simplifies to \(90^{\circ}=m\widehat{PO}-83^{\circ}\). Then, add \(83^{\circ}\) to both sides: \(m\widehat{PO}=90^{\circ} + 83^{\circ}=173^{\circ}\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, the angle outside the circle: the formula is \(\text{measure of angle}=\frac{1}{2}(\text{measure of major arc}-\text{measure of minor arc})\). Wait, maybe the arc \(MO\) is the minor arc, and \(PO\) is the major arc? Wait, no, let's re - examine. Wait, the angle at \(N\) is formed by two secants \(NP\) and \(NO\), intersecting the circle at \(M\) (on \(NP\)) and \(O\) (on \(NO\)). So the formula is \(m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\). Wait, but if we solve \(45=\frac{1}{2}(x - 83)\), then \(x-83 = 90\), \(x=173\)? But that's not matching the options? Wait, no, maybe the arc \(MO\) is \(83^{\circ}\), and we need to find the major arc \(PO\)? Wait, no, the total circumference of a circle is \(360^{\circ}\). Wait, maybe I mixed up the arcs. Wait, let's think again. The angle outside the circle: the measure of the angle is half the difference of the intercepted arcs. The intercepted arcs are the major arc \(PO\) and the minor arc \(MO\). Wait, but if the angle is \(45^{\circ}\), then \(45=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\). So \(m\widehat{PO}=2\times45 + m\widehat{MO}=90 + 83 = 173\)? But the options have 128, 173, 192, 256. Wait, maybe the arc \(MO\) is \(83^{\circ}\), and the angle is formed by a secant and a tangent? Wait, the diagram shows a secant \(NP\) (passing through \(M\) and \(P\)) and a tangent \(NO\) (touching at \(O\))? Wait, if it's a tangent and a secant, then the formula is \(m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\), where \(MO\) is the minor arc, and \(PO\) is the major arc? Wait, no, maybe I got the arcs reversed. Wait, let's check the options. Wait, maybe the angle is \(45^{\circ}\), and the minor arc \(MO\) is \(83^{\circ}\), then the major arc \(PO\) would be \(360 - x\), where \(x\) is the minor arc \(PO\). Wait, no, let's start over.

Wait, the formula for the angle formed outside the circle by two secants (or a secant and a tangent) is: \(m\angle=\frac{1}{2}(\text{measure of the far arc}-\text{measure of the near arc})\). So if the angle at \(N\) is \(45^{\circ}\), the near arc is \(MO = 83^{\circ}\), and the far arc is \(PO\). So \(45=\frac{1}{2}(m\widehat{PO}-83)\). Multiply both sides by 2: \(90=m\widehat{PO}-83\). Then \(m\widehat{PO}=90 + 83=173^{\circ}\). Wait, but the first option was marked as 128, but maybe that's a mistake. Wait, no, maybe I misread the diagram. Wait, maybe the angle is \(45^{\circ}\), and the arc \(MO\) is \(83^{\circ}\), and we need to find the measure of arc \(PO\) (the major arc or the minor arc)? Wait, no, the sum of the minor arc \(PO\) and the major arc \(PO\) is \(360^{\circ}\). Wait, maybe the angle is formed by two secants, and the arc \(MO\) is \(83^{\circ}\), and the angle is \(45^{\circ}\), so the far arc \(PO\) is \(2\times45 + 83=173^{\circ}\). So the measure of arc \(PO\) is \(173^{\circ}\).

Answer:

\(173^{\circ}\) (corresponding to the option: \(173^{\circ}\))