Question

question 7 (10 points)
given ( moverarc{ed} = 44^circ ) and ( mangle bfc = 35^circ ), determine
the measure of the arc ( moverarc{bc} ).
(you may assume that point a is the center of the circle.)
(figure may not be drawn to scale.)

a)

b)

c)

d)

Explanation:

Step1: Recall the tangent-secant angle theorem

The measure of an angle formed by a tangent and a secant (or two secants) outside a circle is half the difference of the measures of the intercepted arcs. For angle \( \angle BFC \), which is formed by secants \( FC \) and \( FE \), the formula is \( m\angle BFC=\frac{1}{2}(m\widehat{BC}-m\widehat{BD}) \).

Step2: Substitute the known values

We know that \( m\angle BFC = 35^\circ \) and \( m\widehat{BD}=44^\circ \). Substitute these into the formula:
\[
35^\circ=\frac{1}{2}(m\widehat{BC}-44^\circ)
\]

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

First, multiply both sides of the equation by 2:
\[
2\times35^\circ=m\widehat{BC}-44^\circ
\]
\[
70^\circ=m\widehat{BC}-44^\circ
\]
Then, add \( 44^\circ \) to both sides:
\[
m\widehat{BC}=70^\circ + 44^\circ
\]
\[
m\widehat{BC}=114^\circ
\]

Answer:

b) \( 114^\circ \)