Question

In circle WW, m, angle, V, W, X, equals, 119, degreesm∠VWX=119
∘
. Solve for xx if m, arc, V, X, equals, left parenthesis, 12, x, plus, 27, right parenthesis, degreesm
VX
⌢
=(12x+27)
∘
. If necessary, round your answer to the nearest tenth.

Answer 1

In a circle, the measure of an angle formed by two chords that intersect inside the circle is equal to half the sum of the measures of the arcs intercepted by the angle. Thus, we can use the following relationship for the angle \( m\angle VWX \) and the arc \( m\overset{\frown}{VX} \):

\[
m\angle VWX = \frac{1}{2} (m \overset{\frown}{VX} + m \overset{\frown}{WY})
\]

However, since we only have the value for \( m\angle VWX \) and the arc \( m\overset{\frown}{VX} \), we will proceed by noting that:

\[
m\angle VWX = \frac{1}{2} m \overset{\frown}{VX}
\]

Given:
- \( m\angle VWX = 119^\circ \)
- \( m \overset{\frown}{VX} = 12x + 27 \)

We can set up the equation:

\[
119 = \frac{1}{2} (12x + 27)
\]

To eliminate the fraction, multiply both sides by 2:

\[
2 \cdot 119 = 12x + 27
\]
\[
238 = 12x + 27
\]

Next, isolate \( 12x \) by subtracting 27 from both sides:

\[
238 - 27 = 12x
\]
\[
211 = 12x
\]

Now, divide by 12 to solve for \( x \):

\[
x = \frac{211}{12}
\]
\[
x \approx 17.5833
\]

Rounding to the nearest tenth:

\[
x \approx 17.6
\]

Thus, the value of \( x \) is:

\[
\boxed{17.6}
\]