Question

question 1 of 10

the length of \\(\widehat{ab}\\) (the minor arc) is 40 cm. what is the circumference of \\(\odot c\\)?

a. 60 cm
b. 240 cm
c. 360 cm
d. 24 cm
e. 6.67 cm
f. 120 cm

Explanation:

Identify given values and target

We are given:

• The length of the minor arc \(\widehat{AB}\) is \(40\text{ cm}\).
• The central angle corresponding to this arc is \(\angle ACB = 60^\circ\).
• We need to find the total circumference of circle \(C\).

Relate arc measure to full circle

Using the Arc Measure and Central Angle concepts, the measure of a minor arc is equal to the measure of its central angle:

$$ m\widehat{AB} = 60^\circ $$

A full circle has a total measure of \(360^\circ\).

Set up the proportion

Using the Minor Arc concept, the length of an arc is a fraction of the total circumference. We set up the ratio:

$$ \frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Arc Measure}}{360^\circ} $$

Substituting the known values:

$$ \frac{40}{\text{Circumference}} = \frac{60^\circ}{360^\circ} $$

Solve for circumference

Simplify the fraction on the right side:

$$ \frac{60^\circ}{360^\circ} = \frac{1}{6} $$

Now, solve the proportion:

$$ \frac{40}{\text{Circumference}} = \frac{1}{6} $$

$$ \text{Circumference} = 40 \times 6 = 240\text{ cm} $$

Answer:

• A. 60 cm
• B. 240 cm (Correct answer)
• C. 360 cm
• D. 24 cm
• E. 6.67 cm
• F. 120 cm