Question

comparing sector areas
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.
(there is a circle with center o, points a, b, c, d on the circumference. oa, ob, oc, od are radii, with ob and oa forming a blue sector aob, oc and od forming a pink sector doc. the length of ob is 5, and the area of sector doc is \\(\frac{4\pi}{9}\\))

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 given circles

Both sectors \( AOB \) and \( DOC \) are in the same circle \( O \), so their radii are equal ( \( OA = OB = OC = OD = 5 \) ). Also, the central angles for sectors \( AOB \) and \( DOC \) are vertical angles (formed by intersecting lines), so they are equal in measure.

Step3: Evaluate each option

• Option 1: Says area of \( AOB \) is greater because central angles are equal. But with equal radii and equal central angles, areas should be equal. So this is wrong.
• Option 2: Says area of \( DOC \) is greater because radii are equal. But equal radii and equal central angles mean equal areas. So this is wrong.
• Option 3: States area of \( AOB \) equals area of \( DOC \) because central angle measures are equal. Since radii are also equal (same circle), using the sector area formula, equal \( \theta \) and equal \( r \) imply equal areas. This is correct.
• Option 4: Says area is equal because radii are equal. But radii alone don't determine sector area; central angle also matters. So this is wrong.

Answer:

The area of sector AOB is equal to the area of sector DOC because the central angle measures are equal. (The corresponding option: The area of sector AOB is equal to the area of sector DOC because the central angle measures are equal.)