Question

which statement is true regarding the two shaded sectors of circle o? explain. the area of sector aob is greater than the area of sector doc because the central angle measures are equal. the area of sector doc is greater than the area of sector aob because the radii are equal. the area of sector aob is equal to the area of sector doc because the central angle measures are equal. the area of sector aob is equal to the area of sector doc because the radii are equal.

Explanation:

Step1: Recall sector area formula

The formula for the area of a sector of a circle is \( A=\frac{\theta}{360^\circ}\times\pi r^2 \) (or \( A = \frac{1}{2}r^2\theta \) when \( \theta \) is in radians), where \( \theta \) is the central angle and \( r \) is the radius of the circle.

Step2: Analyze radii and central angles

• Both sectors \( AOB \) and \( DOC \) are in the same circle \( O \), so their radii (\( OA = OB = OC = OD \)) are equal.
• The central angles for sectors \( AOB \) and \( DOC \) are vertical angles (since \( \angle AOB \) and \( \angle DOC \) are formed by intersecting lines), so they have equal measures (\( \theta_{AOB}=\theta_{DOC} \)).

Step3: Compare sector areas

Using the sector area formula \( A=\frac{\theta}{360^\circ}\times\pi r^2 \), since \( r \) (radius) is the same for both sectors and \( \theta \) (central angle) is the same for both sectors, the area of sector \( AOB \) and sector \( DOC \) must be equal. The key reason is that the central angle measures are equal (and radii are equal, but the equality of central angles is the main factor here as radii are already equal in the same circle).

To determine the true statement, we use the sector area formula \( A=\frac{\theta}{360^\circ}\times\pi r^2 \), where \( \theta \) is the central angle and \( r \) is the radius. Both sectors \( AOB \) and \( DOC \) lie in circle \( O \), so their radii are equal. The central angles \( \angle AOB \) and \( \angle DOC \) are vertical angles, so they have equal measures. Since \( r \) and \( \theta \) are equal for both sectors, their areas are equal. Among the options, the statement "The area of sector AOB is equal to the area of sector DOC because the central angle measures are equal" is true.

Answer:

The area of sector AOB is equal to the area of sector DOC because the central angle measures are equal.