Question

question 8 (10 points) given ( mangle dbe = 70^circ ) and ( moverarc{dfb} = (9x + 4)^circ ), determine the most appropriate value for ( x ). (you may assume that point ( a ) is the center of the circle.)
a) ( x = 12 )
b) ( x = 24 )
c) ( x = 7.3 )
d) ( x = 15.1 )

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. Here, \( \angle DBE \) is formed by tangent \( BE \) and chord \( BD \), so \( m\angle DBE=\frac{1}{2}m\widehat{DFB} \).

Step2: Substitute the given values

We know \( m\angle DBE = 70^\circ \) and \( m\widehat{DFB}=(9x + 4)^\circ \). Substituting into the formula:
\( 70=\frac{1}{2}(9x + 4) \)

Step3: Solve for \( x \)

Multiply both sides by 2:
\( 140=9x + 4 \)

Subtract 4 from both sides:
\( 136 = 9x \)

Divide both sides by 9:
\( x=\frac{136}{9}\approx15.1 \)? Wait, no, wait—wait, maybe I misread the arc. Wait, if \( BE \) is a tangent, and \( \angle DBE \) is 70°, then the intercepted arc \( \widehat{DB} \) would be 140°, but the arc \( \widehat{DFB} \) is the major arc? Wait, no, maybe the diagram: point \( A \) is the center, so \( BE \) is a tangent (since it's a straight line at \( B \), tangent to the circle). So the angle between tangent \( BE \) and chord \( BD \) is equal to half the measure of the intercepted arc \( \widehat{DFB} \)? Wait, no, the intercepted arc should be the arc opposite the angle, the one not containing the tangent's side. Wait, actually, the correct formula is that the measure of the angle between tangent and chord is half the measure of the intercepted arc. So if \( \angle DBE = 70^\circ \), then the intercepted arc \( \widehat{DFB} \) (the arc that's "cut off" by the angle, not including the tangent side) should be \( 2\times70 = 140^\circ \)? Wait, no, wait: no, the angle formed by tangent and chord is half the measure of its intercepted arc. So \( m\angle DBE=\frac{1}{2}m\widehat{DFB} \), so \( 70=\frac{1}{2}(9x + 4) \). Wait, solving that:

\( 70\times2=9x + 4 \)
\( 140 = 9x + 4 \)
\( 9x=136 \)
\( x=\frac{136}{9}\approx15.1 \), but the options have \( x = 15.1 \) as option d? Wait, but let's check the options again. Wait, maybe I made a mistake. Wait, maybe the arc is \( \widehat{DB} \), but no, the problem says \( m\widehat{DFB}=(9x + 4)^\circ \). Wait, maybe the diagram: \( F \) is another point, so \( \widehat{DFB} \) is the arc from \( D \) to \( F \) to \( B \), which is the major arc? Wait, no, if \( A \) is the center, and \( BE \) is tangent, then \( \angle DBE = 70^\circ \), so the inscribed angle or the tangent-chord angle: tangent-chord angle is half the intercepted arc. So the intercepted arc is \( \widehat{DB} \), but if \( \widehat{DFB} \) is the rest of the circle, then \( m\widehat{DFB}=360 - m\widehat{DB} \). Wait, no, maybe the angle is formed by a secant, but no, \( BE \) is tangent. Wait, let's re-express:

Tangent-chord angle theorem: \( m\angle = \frac{1}{2}m\text{(intercepted arc)} \). So \( \angle DBE = 70^\circ \), so intercepted arc \( \widehat{DB} \) (the minor arc) would be \( 140^\circ \), but if \( \widehat{DFB} \) is the major arc, then \( m\widehat{DFB}=360 - 140 = 220^\circ \), but that doesn't match. Wait, maybe the diagram has \( BE \) as a tangent, and \( BD \) as a chord, and \( \widehat{DFB} \) is the arc that's equal to \( 2\times(180 - 70) \)? No, wait, maybe I messed up. Wait, the options: a) 12, b)24, c)7.3, d)15.1. Let's check option b: if \( x = 24 \), then \( 9x + 4 = 220 \), half of 220 is 110, not 70. Option a: \( 9\times12 + 4 = 112 \), half is 56, no. Option c: \( 9\times7.3 + 4 \approx 69.7 \), half is ~34.8, no. Option d: \( 9\times15.1 + 4 \approx 139.9 \), half is ~70, which matches. So that's correct. Wait, so my initial calculation was right: \( 70 = \frac{1}{2}(9x + 4) \), so \( 140 = 9x + 4 \), \( 9x = 136 \), \( x = 136/9 \approx 15.1 \), which is option d.

Answer:

d) \( x = 15.1 \)