Question

question 36 of 44 what is the measure of (widehat{ed}) (the minor arc) in the diagram below? diagram of a circle with tangent at e, chord/secant ed, and angle 78° at e between tangent and ed click here for long description
a. 156°
b. 39°
c. 204°
d. 78°

Explanation:

Step1: Recall the tangent-chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Let the measure of arc \( ED \) be \( x \). The angle between the tangent and chord \( EF \) and \( ED \) is \( 78^\circ \), so by the theorem, \( 78^\circ=\frac{1}{2}x \)? Wait, no, wait. Wait, actually, the tangent is perpendicular to the radius, but here, the angle between tangent \( E \) (the tangent line at \( E \)) and chord \( ED \) is given as \( 78^\circ \). Wait, no, the correct theorem: the measure of the angle between a tangent and a chord is equal to half the measure of the intercepted arc. Wait, but if the angle is \( 78^\circ \), then the intercepted arc \( ED \) (the minor arc) should be twice that? Wait, no, wait. Wait, no, the angle between tangent and chord is half the intercepted arc. Wait, let's correct. Let me re-express. The tangent at \( E \), so the radius \( OE \) (if \( O \) is the center) is perpendicular to the tangent. But in the diagram, we have a chord \( ED \) and a tangent at \( E \), with the angle between tangent and chord \( ED \) being \( 78^\circ \). Wait, actually, the measure of the angle between tangent and chord is equal to half the measure of the intercepted arc. So if the angle is \( 78^\circ \), then the intercepted arc \( ED \) (the minor arc) is \( 2\times78^\circ = 156^\circ \)? Wait, no, wait, no. Wait, no, the angle between tangent and chord is half the intercepted arc. Wait, let's think again. The tangent at \( E \), so the angle between tangent and chord \( ED \) is equal to half the measure of the arc \( ED \) that is opposite to the angle. Wait, no, the formula is: the measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc. So if the angle is \( 78^\circ \), then the intercepted arc \( ED \) (the arc that is "cut off" by the chord \( ED \) and the tangent) has measure \( 2\times78^\circ = 156^\circ \)? Wait, but let's check the options. Option A is \( 156^\circ \), B is \( 39^\circ \), C is \( 204^\circ \), D is \( 78^\circ \). Wait, maybe I got the theorem reversed. Wait, no, the angle between tangent and chord is half the intercepted arc. So if the angle is \( 78^\circ \), then the arc \( ED \) (minor arc) is \( 2\times78 = 156^\circ \). Let's confirm. The tangent at \( E \), chord \( ED \). The angle between tangent and chord is equal to half the measure of the intercepted arc \( ED \). So angle \( = \frac{1}{2} \times \) arc \( ED \). Therefore, arc \( ED = 2 \times \) angle. So \( 2 \times 78^\circ = 156^\circ \). So the measure of minor arc \( ED \) is \( 156^\circ \).

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

Using the tangent - chord angle theorem: \( m\angle FED=\frac{1}{2}m\widehat{ED} \) (wait, no, actually, the angle between tangent and chord is equal to half the measure of the intercepted arc. So if \( \angle FED = 78^\circ \), then \( 78^\circ=\frac{1}{2}m\widehat{ED} \)? No, that would give \( m\widehat{ED}=156^\circ \), which is option A. Wait, yes, that's correct. So \( m\widehat{ED} = 2\times78^\circ=156^\circ \).

Answer:

A. 156°