AMC 8 Daily Practice Round 11

Complete problem set with solutions and individual problem pages

Problem 21 Medium

As shown in the figure, a circular piece of paper is cut into two sectors, \alpha and \beta. The area of \alpha is 18, and the central angle of \alpha is 72^\circ larger than that of \beta. What is the area of \beta?

  • A.

    16

  • B.

    12

  • C.

    14

  • D.

    10

  • E.

    8

Answer:B

Since the two sectors have the same radius, the ratio of their areas is equal to the ratio of their central angles. Given that the difference in their central angles is n_\alpha - n_\beta = 72^\circ and the sum of their central angles is 360^\circ. We can solve for n_\alpha = 216^\circ and n_\beta =144^\circ. Hence, the ratio of their central angles is 216:144 = 3:2. It follows that the area ratio is also 3:2. Thus, the area of \beta is 12.

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