Question

consider the following two sectors.
which statement correctly gives the unknown arc length and the justincation for the measurement?
a. the unknown arc length is 37 cm. because the radius is 5 units shorter in length, the arc must be 5 units shorter in length.

b. the unknown arc length is 58.15 cm. because arc lengths are inversely proportional to radius lengths, the smaller circle must have a longer arc.

c. the unknown arc length is 30.33 cm. because all circles are similar, the length of the arc intercepted by an angle is proportional to the radius.

d. the unknown arc length is 42 cm. because all circles are congruent, congruent angles must intercept congruent arcs.

Explanation:

Step1: Set up proportion

Since arc length $l$ is proportional to radius $r$ for equal central angles, we use:
$\frac{l_1}{r_1} = \frac{l_2}{r_2}$
Here, $l_1=42$ cm, $r_1=18$ cm, $r_2=13$ cm, $l_2$ is unknown.

Step2: Solve for unknown arc length

Rearrange to solve for $l_2$:
$l_2 = \frac{l_1 \times r_2}{r_1}$
Substitute values:
$l_2 = \frac{42 \times 13}{18} = \frac{546}{18} \approx 30.33$ cm

Step3: Verify reasoning

Circles are similar, so arc length and radius are directly proportional for congruent central angles, matching option C's logic.

Answer:

C. The unknown arc length is 30.33 cm. Because all circles are similar, the length of the arc intercepted by an angle is proportional to the radius.