Question

using portions of a circle
what is the measure of the central angle?
°
ratio represents the measure of the central angle
red to the measure of the entire circle?
\\(\frac{\theta}{360^{\circ}}(2\pi r)\\), what is the length of minor arc ab?
circle with center c, radius cb = 18, central angle ∠acb = 140°
options for central angle: 90, 140, 180, 360

Explanation:

First Question: Measure of the central angle

From the diagram, the central angle ∠ACB is labeled as \( 140^\circ \). So the measure of the central angle is \( 140^\circ \).

The measure of the entire circle (a full - rotation) is \( 360^\circ \). The central angle is \( 140^\circ \). The ratio of the central angle to the entire circle is \( \frac{140}{360} \), so the numerator (the measure of the central angle) is \( 140 \) and the denominator (the measure of the entire circle) is \( 360 \). So the ratio representing the measure of the central angle compared to the measure of the entire circle has the central angle measure as \( 140 \).

Step 1: Recall the arc - length formula

The formula for the length of an arc \( s \) is \( s=\frac{\theta}{360^\circ}\times(2\pi r) \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step 2: Identify the values of \( \theta \) and \( r \)

From the diagram, \( \theta = 140^\circ \) and \( r = 18 \).

Step 3: Substitute the values into the formula

Substitute \( \theta=140^\circ \) and \( r = 18 \) into the formula \( s=\frac{\theta}{360^\circ}\times(2\pi r) \).
\[

$$\begin{align*} s&=\frac{140^\circ}{360^\circ}\times(2\pi\times18)\\ &=\frac{140}{360}\times36\pi\\ &=\frac{140\times36\pi}{360}\\ & = 14\pi \end{align*}$$

\]
If we want a decimal approximation (using \( \pi\approx3.14 \)):
\[
s=14\times3.14 = 43.96
\]

Answer:

\( 140^\circ \)

Second Question: Ratio of central angle to entire circle