Question

1. in the diagram below, circle o has a radius of 8 cm and a central angle that measures ( 140^circ ). what is the length of ( overarc{ab} ), to the nearest centimeter?

(1) 20
(2) 10
(3) 78
(4) 25

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( s=\frac{\theta}{360^{\circ}}\times2\pi r \).

Step2: Identify the values

Here, \( r = 8\space\text{cm} \) and \( \theta=140^{\circ} \).

Step3: Substitute the values into the formula

Substitute \( r = 8 \) and \( \theta = 140^{\circ} \) into the formula:
\( s=\frac{140^{\circ}}{360^{\circ}}\times2\pi\times8 \)
First, simplify \( \frac{140}{360}=\frac{7}{18} \). Then, \( 2\times8 = 16 \). So the formula becomes \( s=\frac{7}{18}\times16\pi \).
Calculate \( \frac{7\times16}{18}\pi=\frac{112}{18}\pi=\frac{56}{9}\pi \approx\frac{56}{9}\times3.1416 \)
\( \frac{56\times3.1416}{9}\approx\frac{175.9296}{9}\approx19.5477\approx20 \) (to the nearest centimeter)

Answer:

20