Question

the circle above with center \\(o\\) has a circumference of 36. what is the length of minor arc \\(\widehat{ac}\\)?

a) 9
b) 12
c) 18
d) 36

Explanation:

🆕 New Concept Discovered: Arc Length and Central Angles
An arc's length is proportional to its central angle.

Step 1: Identify the central angle

The angle symbol at the center \(O\) indicates that the central angle \(\angle AOC\) is a right angle:

$$ \angle AOC = 90^\circ $$

Step 2: Determine the fraction of the circle

A full circle has a total measure of \(360^\circ\). We find the fraction of the circle that the minor arc \(\widehat{AC}\) represents:

$$ \text{Fraction} = \frac{90^\circ}{360^\circ} = \frac{1}{4} $$

Step 3: Calculate the arc length

Since the arc is \(\frac{1}{4}\) of the entire circle, its length is \(\frac{1}{4}\) of the total circumference:

$$ \text{Length of } \widehat{AC} = \frac{1}{4} \times 36 = 9 $$

Answer:

A) 9