Question

drag each tile to the correct box. consider the given circles. order the sectors from least to greatest according to their areas, in square units. sector a, sector c, sector b tiles and circle diagrams with sector a (radius 4 units, angle 169°), sector b (radius 9 units, angle 82°), sector c (radius 6 units, angle 117°)

Explanation:

Step1: Recall Sector Area Formula

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

Step2: Calculate Area of Sector A

For Sector A: \( \theta = 169^\circ \), \( r = 4 \) units.
\( A_A = \frac{169}{360} \times \pi \times 4^2 = \frac{169}{360} \times 16\pi \approx \frac{169 \times 16}{360}\pi \approx \frac{2704}{360}\pi \approx 7.51\pi \).

Step3: Calculate Area of Sector B

For Sector B: \( \theta = 82^\circ \), \( r = 9 \) units.
\( A_B = \frac{82}{360} \times \pi \times 9^2 = \frac{82}{360} \times 81\pi = \frac{82 \times 81}{360}\pi = \frac{6642}{360}\pi \approx 18.45\pi \).

Step4: Calculate Area of Sector C

For Sector C: \( \theta = 117^\circ \), \( r = 6 \) units.
\( A_C = \frac{117}{360} \times \pi \times 6^2 = \frac{117}{360} \times 36\pi = \frac{117 \times 36}{360}\pi = \frac{4212}{360}\pi = 11.7\pi \).

Step5: Compare the Areas

Now compare \( A_A \approx 7.51\pi \), \( A_C = 11.7\pi \), \( A_B \approx 18.45\pi \).
So, \( A_A < A_C < A_B \).

Answer:

Sector A, Sector C, Sector B