Question

answer the following.
when applicable, give the exact answer in terms of π in simplest form.
(a) for each circle, find the length of the arc that is intercepted by the central angle.
length of \\(\overarc{ab}\\): \\(\square\\) cm
length of \\(\overarc{de}\\): \\(\square\\) cm

Explanation:

Step1: Find length of arc \( \overarc{AB} \)

The formula for the length of an arc with central angle \( \theta \) (in radians) and radius \( r \) is \( s = r\theta \). For a semicircle, the central angle \( \theta = \pi \) radians. The radius of the circle with center \( C \) is \( r = 2 \) cm. So, arc length \( \overarc{AB} = 2 \times \pi = 2\pi \) cm? Wait, no, wait. Wait, the central angle for \( AB \): looking at the diagram, \( AB \) is a semicircle? Wait, \( CB \) and \( CA \) are radii, and the angle between them: if \( AB \) is a semicircle, the central angle is \( \pi \) (180 degrees). Wait, but the radius is 2 cm. Wait, the formula for arc length is also \( \frac{\theta}{2\pi} \times 2\pi r = r\theta \) (when \( \theta \) is in radians). For a semicircle, \( \theta = \pi \), so arc length is \( r\pi \). Wait, radius \( r = 2 \) cm, so arc length \( AB \): wait, maybe the central angle is \( \pi \) (since \( AB \) is a semicircle? Wait, the diagram shows \( C \) as center, \( A \) and \( B \) on the circle, with \( CB \) vertical and \( CA \) horizontal? Wait, no, maybe the central angle is \( \pi \) (180 degrees). Wait, no, maybe I misread. Wait, the first circle: center \( C \), radius \( 2 \) cm, points \( A \) and \( B \). Wait, maybe the central angle for \( AB \) is \( \pi \) (semicircle). So arc length \( AB = r\theta = 2 \times \pi = 2\pi \)? Wait, no, wait, the formula for arc length is \( \frac{\theta}{360^\circ} \times 2\pi r \). If the central angle is \( 180^\circ \) (semicircle), then \( \frac{180^\circ}{360^\circ} \times 2\pi \times 2 = \frac{1}{2} \times 4\pi = 2\pi \) cm. Wait, but maybe the central angle is \( \pi \) radians, so \( s = r\theta = 2 \times \pi = 2\pi \) cm.

Step2: Find length of arc \( \overarc{DE} \)

The circle with center \( F \) has radius \( r = 1 \) cm. The central angle for \( DE \): looking at the diagram, \( DE \) is a semicircle? Wait, \( FE \) and \( FD \) are radii, and the angle between them: if \( DE \) is a semicircle, central angle \( \theta = \pi \) radians. So arc length \( \overarc{DE} = 1 \times \pi = \pi \) cm? Wait, no, wait. Wait, the formula: arc length \( s = \frac{\theta}{2\pi} \times 2\pi r = r\theta \). For a semicircle, \( \theta = \pi \), so \( s = r\pi \). Radius \( r = 1 \) cm, so \( s = 1 \times \pi = \pi \) cm. Wait, but maybe the central angle is \( \pi \) (180 degrees), so \( \frac{180^\circ}{360^\circ} \times 2\pi \times 1 = \pi \) cm.

Wait, let's recheck. For arc \( AB \): center \( C \), radius \( 2 \) cm. The central angle: if \( AB \) is a semicircle (since \( CB \) and \( CA \) are radii forming a straight line? Wait, the diagram shows \( C \) with \( CA \) and \( CB \) as radii, and \( AB \) as the arc. Wait, maybe the central angle is \( \pi \) (180 degrees). So arc length \( AB = \frac{\pi}{2\pi} \times 2\pi \times 2 = \frac{1}{2} \times 4\pi = 2\pi \) cm. For arc \( DE \): center \( F \), radius \( 1 \) cm. Central angle is \( \pi \) (semicircle), so arc length \( DE = \frac{\pi}{2\pi} \times 2\pi \times 1 = \pi \) cm. Wait, but maybe I made a mistake. Wait, the problem says "the arc that is intercepted by the central angle". Wait, maybe the central angle for \( AB \) is \( \pi \) (180 degrees), so arc length \( AB = r\theta = 2 \times \pi = 2\pi \) cm? Wait, no, \( \theta \) in radians: \( \pi \) radians, so \( s = r\theta = 2 \times \pi = 2\pi \) cm. For \( DE \), radius \( 1 \) cm, central angle \( \pi \) radians, so \( s = 1 \times \pi = \pi \) cm. Wait, but let's confirm the formula. Arc length formula: \( s = r\theta \), where \( \theta \) is the central angle in radians. For a semicircle, \( \theta = \pi \), so yes. So:

Length of \( \overarc{AB} \): radius \( r = 2 \) cm, \( \theta = \pi \), so \( s = 2 \times \pi = 2\pi \) cm? Wait, no, wait, \( 2\pi r \) is the circumference. A semicircle is half the circumference, so \( \frac{1}{2} \times 2\pi r = \pi r \). So for \( r = 2 \), semicircle length is \( 2\pi \) cm? Wait, \( \pi r = 2\pi \), yes. For \( r = 1 \), semicircle length is \( \pi \times 1 = \pi \) cm. Yes, that's correct. So:

Step1 (Arc AB):

Circumference of circle with \( r = 2 \) is \( 2\pi r = 4\pi \). Arc \( AB \) is a semicircle (central angle \( 180^\circ \), which is \( \frac{1}{2} \) of the circle), so length is \( \frac{1}{2} \times 4\pi = 2\pi \) cm.

Step2 (Arc DE):

Circumference of circle with \( r = 1 \) is \( 2\pi r = 2\pi \). Arc \( DE \) is a semicircle (central angle \( 180^\circ \), \( \frac{1}{2} \) of the circle), so length is \( \frac{1}{2} \times 2\pi = \pi \) cm.

Answer:

Length of \( \overarc{AB} \): \( 2\pi \) cm
Length of \( \overarc{DE} \): \( \pi \) cm