Question

secants, tangents, and angles
what is ( moverarc{po} )?
128°
256°
173°
192°

Explanation:

Step1: Recall the formula for angle formed by secant and tangent

The measure of an angle formed by a secant and a tangent outside a circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO}) \).

Step2: Substitute known values

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

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

Multiply both sides by 2: \( 45\times2=m\widehat{PO}-83 \), so \( 90 = m\widehat{PO}-83 \). Then add 83 to both sides: \( m\widehat{PO}=90 + 83=173^{\circ} \)? Wait, no, wait. Wait, maybe I mixed up the arcs. Wait, the total circumference arc is \( 360^{\circ} \), but no, the angle outside is half the difference of the major arc and the minor arc. Wait, maybe I made a mistake. Wait, let's re - check. The formula is \( m\angle=\frac{1}{2}(\text{major arc}-\text{minor arc}) \). Wait, if the angle is outside, then it's half the difference of the intercepted arcs, where the major arc is \( \widehat{PO} \) and the minor arc is \( \widehat{MO} \)? Wait, no, the two intercepted arcs are the major arc \( \widehat{PO} \) and the minor arc \( \widehat{MO} \)? Wait, no, the secant and tangent intercept two arcs: the major arc \( \widehat{PO} \) and the minor arc \( \widehat{MO} \)? Wait, no, the points are P, M, O on the circle. So the secant is NP (passing through M) and the tangent is NO. So the intercepted arcs are \( \widehat{PO} \) (major) and \( \widehat{MO} \) (minor). So the formula is \( m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO}) \). Wait, but when we solved \( 45=\frac{1}{2}(x - 83) \), we got \( x=45\times2 + 83=173 \), but let's check the options. Wait, but maybe I had the formula reversed. Wait, no, the angle formed outside is half the difference of the measures of the intercepted arcs, with the larger arc minus the smaller arc. Wait, but let's check the options. The options are 128, 256, 173, 192. Wait, maybe I made a mistake in the arc labels. Wait, maybe the minor arc is \( \widehat{PMO} \)? No, the diagram shows arc MO is 83 degrees. Wait, let's recast. Let \( x = m\widehat{PO} \). Then the formula is \( 45=\frac{1}{2}(x - 83) \), so \( x=45\times2+83 = 173 \). But wait, another way: the sum of the minor arc and the major arc is \( 360^{\circ} \), but no, the two arcs intercepted by the secant and tangent are the major arc \( \widehat{PO} \) and the minor arc \( \widehat{MO} \), and their difference is related to the angle. Wait, maybe the correct formula is \( m\angle=\frac{1}{2}(\text{major arc}-\text{minor arc}) \). So if we solve \( 45=\frac{1}{2}(x - 83) \), we get \( x = 45\times2+83=173 \). But let's check the options. 173 is one of the options. Wait, but let's verify again.

Wait, maybe I had the formula wrong. Let's recall: The measure of an angle formed by a secant and a tangent drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc and the minor arc that lie between the secant and the tangent. So if the angle at N is \( 45^{\circ} \), and the minor arc \( \widehat{MO} \) is \( 83^{\circ} \), then:

\( m\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO}) \)

Substitute \( m\angle N = 45^{\circ} \) and \( m\widehat{MO}=83^{\circ} \):

\( 45=\frac{1}{2}(m\widehat{PO}-83) \)

Multiply both sides by 2:

\( 90=m\widehat{PO}-83 \)

Add 83 to both sides:

\( m\widehat{PO}=90 + 83=173^{\circ} \)

Answer:

\( 173^{\circ} \) (the option with \( 173^{\circ} \))