Question

the ( mangle feb = 78^circ ), find the measure of arc ( eab ).
options: ( 78^circ ), ( 360^circ ), ( 204^circ ), ( 156^circ )

Explanation:

Step1: Identify tangent and secant properties

Line \( EF \) is a tangent at \( E \), and \( EB \) is a secant. The measure of the angle between tangent and secant (\( \angle FEB \)) is half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc \( EAB \) and the minor arc \( EB \). Also, the measure of a tangent - secant angle \( \theta=\frac{1}{2}(m\overset{\frown}{EAB}-m\overset{\frown}{EB}) \). But we also know that the angle between a tangent and a secant: the measure of \( \angle FEB \) is half the measure of the intercepted major arc \( EAB \) if we consider the linear pair and the fact that the tangent is perpendicular to the radius, but more accurately, the formula for the angle formed by a tangent and a secant is \( m\angle FEB=\frac{1}{2}(m\overset{\frown}{EAB}) \) when we consider the arc opposite to the angle? Wait, no. Wait, the correct formula is: the measure of an angle formed by a tangent and a secant drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs. But in this case, point \( E \) is on the circle (since \( EF \) is tangent at \( E \) and \( EB \) is a chord). Wait, when the point is on the circle, the angle between tangent and chord is equal to half the measure of the intercepted arc. So \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EB} \)? No, wait, the angle between a tangent and a chord is equal to half the measure of the intercepted arc. So if \( EF \) is tangent at \( E \) and \( EB \) is a chord, then \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EB} \). Wait, no, actually, the angle between tangent and chord is equal to the measure of the inscribed angle on the opposite side of the chord. So \( m\angle FEB = \frac{1}{2}m\overset{\frown}{EB} \)? Wait, let's recall: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if \( \angle FEB \) is formed by tangent \( EF \) and chord \( EB \), then \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EB} \). So \( 78^{\circ}=\frac{1}{2}m\overset{\frown}{EB} \), then \( m\overset{\frown}{EB} = 156^{\circ} \)? No, that can't be. Wait, no, when the angle is formed by a tangent and a chord at the point of tangency, the measure of the angle is equal to half the measure of the intercepted arc. So \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EAB} \)? Wait, no, let's draw this mentally. The tangent \( EF \) at \( E \), chord \( EB \). The arc \( EAB \) and arc \( EB \) are related. The total circumference is \( 360^{\circ} \). The angle on a straight line at \( E \) (since \( EF \) and the line through \( E \) opposite to \( EF \) form a straight line, \( 180^{\circ} \)). Wait, the line \( E \) with the tangent \( EF \) and the chord \( EB \): the angle between tangent \( EF \) and chord \( EB \) is \( 78^{\circ} \), so the angle between the tangent and the other side (the arc \( EAB \)): Wait, the sum of the measure of arc \( EAB \) and arc \( EB \) is \( 360^{\circ} \)? No, arc \( EAB \) and arc \( EB \): arc \( EB \) is a minor arc, arc \( EAB \) is a major arc? Wait, no, \( E \), \( A \), \( B \) are on the circle. So arc \( EAB \) and arc \( EB \): arc \( EB \) is from \( E \) to \( B \), arc \( EAB \) is from \( E \) to \( A \) to \( B \). The measure of the angle between tangent \( EF \) and chord \( EB \) is equal to half the measure of the intercepted arc \( EAB \)? Wait, no, let's use the correct theorem: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \( \angle FEB \), the intercepted arc is \( \overset{\frown}{EB} \)? No, that would mean \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EB} \), but then the angle between tangent and chord is equal to the inscribed angle over the same arc. Wait, maybe I got it reversed. Let's think: If we have a tangent at \( E \) and a chord \( EB \), then the angle between tangent \( EF \) and chord \( EB \) ( \( \angle FEB \)) is equal to the measure of the inscribed angle that intercepts arc \( EB \). So \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EB} \). Then \( m\overset{\frown}{EB} = 2\times78^{\circ}=156^{\circ} \)? But that can't be, because the total circumference is \( 360^{\circ} \), and arc \( EAB \) would be \( 360 - 156=204^{\circ} \)? Wait, no, wait. Wait, the angle between tangent and chord is equal to half the measure of the intercepted arc. So if the angle is \( 78^{\circ} \), the intercepted arc (the arc that is "cut off" by the chord \( EB \) and the tangent \( EF \)) is arc \( EAB \)? No, I think I made a mistake. Let's recall the correct theorem: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. The intercepted arc is the arc that is inside the angle. Wait, when the angle is at the point of tangency, the angle between tangent and chord is equal to half the measure of the intercepted arc (the arc that is opposite to the angle, i.e., the arc that is not adjacent to the angle). So if \( \angle FEB \) is at \( E \), with tangent \( EF \) and chord \( EB \), then the intercepted arc is arc \( EAB \), and \( m\angle FEB=\frac{1}{2}m\overset{\frown}{EAB} \)? No, that would mean \( m\overset{\frown}{EAB}=2\times78 = 156^{\circ} \), but that's not right. Wait, no, the correct formula is: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \( \angle FEB \), the intercepted arc is arc \( EB \). Wait, let's take an example: if the angle is \( 90^{\circ} \), the arc would be \( 180^{\circ} \), which is a semicircle. So if the angle is \( 78^{\circ} \), the arc \( EB \) would be \( 156^{\circ} \), but then the remaining arc \( EAB \) would be \( 360 - 156 = 204^{\circ} \). Wait, but let's check the answer options. The options are \( 78^{\circ},360^{\circ},204^{\circ},156^{\circ} \). Wait, maybe the angle between tangent and chord is equal to half the measure of the major arc. Wait, no, let's re - derive.

Let’s denote: Let \( O \) be the center of the circle ( \( D \) in the diagram). The radius \( DE \) is perpendicular to the tangent \( EF \), so \( \angle DEF = 90^{\circ} \). We know that \( \angle FEB=78^{\circ} \), so \( \angle DEB=90 - 78 = 12^{\circ} \). Triangle \( DEB \) is isosceles ( \( DE = DB \), radii), so \( \angle DBE=\angle DEB = 12^{\circ} \), then the central angle \( \angle EDB=180-(12 + 12)=156^{\circ} \), which is the measure of arc \( EB \) (since central angle equals arc measure). Then the measure of arc \( EAB \) is \( 360 - 156 = 204^{\circ} \).

Wait, another way: The angle between tangent and chord is equal to half the measure of the intercepted arc. The intercepted arc for \( \angle FEB \) is arc \( EAB \)? No, wait, the angle between tangent and chord is equal to half the measure of the intercepted arc. The intercepted arc is the arc that is not adjacent to the angle. So if \( \angle FEB \) is at \( E \), the adjacent arc is \( EB \), and the non - adjacent arc is \( EAB \). Then \( m\angle FEB=\frac{1}{2}(m\overset{\frown}{EAB}-m\overset{\frown}{EB}) \)? No, when the point is on the circle, the angle between tangent and chord is equal to half the measure of the intercepted arc (the arc that is "cut off" by the chord and lies in the alternate segment). So the alternate segment theorem: the angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment. So \( \angle FEB=\angle EAB \), and \( \angle EAB \) is an inscribed angle intercepting arc \( EB \), so \( \angle EAB=\frac{1}{2}m\overset{\frown}{EB} \), which would mean \( m\overset{\frown}{EB}=2\times78 = 156^{\circ} \), and then arc \( EAB = 360 - 156 = 204^{\circ} \).

Step2: Calculate the measure of arc \( EAB \)

We know that the total measure of a circle is \( 360^{\circ} \). Let \( m\overset{\frown}{EB}=x \). From the alternate segment theorem (angle between tangent and chord equals half the measure of the intercepted arc), we have \( m\angle FEB=\frac{1}{2}x \). Given \( m\angle FEB = 78^{\circ} \), then \( 78^{\circ}=\frac{1}{2}x \), so \( x = 156^{\circ} \) (measure of arc \( EB \)).

Then, the measure of arc \( EAB \) is \( 360^{\circ}-m\overset{\frown}{EB} \). Substituting \( m\overset{\frown}{EB} = 156^{\circ} \), we get \( m\overset{\frown}{EAB}=360 - 156=204^{\circ} \).

Answer:

\( 204^{\circ} \) (corresponding to the option with \( 204^{\circ} \))