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 the formula for the area of a sector

The area of a sector of a circle is given by \(A=\frac{1}{2}r^{2}\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central - angle (in radians) of the sector.

Step2: Analyze the relationship between \(\angle AOB\) and \(\angle DOC\)

Since \(\angle AOB\) and \(\angle DOC\) are vertical angles. By the vertical - angles theorem, \(\angle AOB=\angle DOC\). Let \(\theta=\angle AOB = \angle DOC\).

Step3: Analyze the radii of the two sectors

The radius of sector \(AOB\) is \(r_{1}=OA = 5\), and the radius of sector \(DOC\) is \(r_{2}=OC\). Since \(OA\) and \(OC\) are radii of the same circle \(O\), \(r_{1}=r_{2}=5\).

Step4: Calculate the area of sector \(AOB\)

Using the formula \(A=\frac{1}{2}r^{2}\theta\), for sector \(AOB\) with \(r = 5\), \(A_{AOB}=\frac{1}{2}\times5^{2}\times\theta=\frac{25}{2}\theta\).

Step5: Calculate the area of sector \(DOC\)

Using the formula \(A=\frac{1}{2}r^{2}\theta\), for sector \(DOC\) with \(r = 5\), \(A_{DOC}=\frac{1}{2}\times5^{2}\times\theta=\frac{25}{2}\theta\).

Answer:

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