Question

(c) choose the equation that is true.
area of sector = \frac{360^{\circ}}{m\overarc{ab}} \cdot area of circle
area of sector = \frac{m\overarc{ab}}{360^{\circ}} \cdot area of circle
area of sector = \frac{m\overarc{ab}}{360^{\circ}}
area of sector = m\overarc{ab}
(d) using \pi r^2 for the area of the circle, choose the equation that gives a formula for the area of the sector.
area of sector = \frac{360^{\circ}}{m\overarc{ab}} \cdot \pi r^2
area of sector = \frac{m\overarc{ab}}{360^{\circ}} \cdot \pi r^2
area of sector = m\overarc{ab} \cdot \pi r^2
area of sector = \frac{m\overarc{ab}}{360^{\circ}} \cdot 8\pi r^2

Explanation:

Step1: Recall Sector Area Formula

The area of a sector is a fraction of the area of the whole circle. The fraction is determined by the central angle \( m\overarc{AB} \) (in degrees) compared to the total degrees in a circle (\( 360^\circ \)). The area of a circle is \( \pi r^2 \). So the area of the sector should be \( \frac{m\overarc{AB}}{360^\circ} \times \pi r^2 \), which is \( \text{Area of sector} = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \).

Step2: Match with Options

Looking at the options for part (d), the correct formula is \( \text{Area of sector} = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \), which corresponds to the option "Area of sector \( = \frac{360^\circ}{m\overarc{AB}} \cdot \pi r^2 \)"? Wait, no, rechecking: the correct formula is \( \text{Area of sector} = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \), so among the options, the one with \( \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \) (the second option in the top - left of part (d) maybe? Wait, the options for (d) are:

1. Area of sector \( = \frac{360^\circ}{m\overarc{AB}} \cdot \pi r^2 \) (incorrect, fraction is angle over 360, not 360 over angle)
2. Area of sector \( = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \) (correct)
3. Area of sector \( = m\overarc{AB} \cdot \pi r^2 \) (incorrect, no division by 360)
4. Area of sector \( = \frac{m\overarc{AB}}{360^\circ} \cdot 8\pi r^2 \) (incorrect, 8 is not part of the standard formula)

So the correct option is the second one in the list for part (d), which is "Area of sector \( = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \)".

Answer:

For part (d), the correct option is "Area of sector \( = \frac{m\overarc{AB}}{360^\circ} \cdot \pi r^2 \)" (the second option in the set of options for part (d)).