Multiple choice

The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

  1. False, since it requires the condition $2{ { r }_{ 1 } }^{ 2 }<{ { r }_{ 2 } }^{ 2 }\\ $
  2. True, since given that $2{ { r }_{ 1 } }^{ 2 }={ { r }_{ 2 } }^{ 2 }.\\ $
  3. False, since it requires the condition $2{ { r }_{ 1 } }^{ 2 }={ { r }_{ 2 } }^{ 2 }.\\ $
  4. True, since given that $2{ { r }_{ 1 } }^{ 2 }>{ { r }_{ 2 } }^{ 2 }.\\ $
Reveal answer Fill a bubble to check yourself
C Correct answer
AI explanation

The area of a sector is (theta/360) times pi times r squared, while the arc length is (theta/360) times 2 times pi times r. Equal areas do not guarantee equal arc lengths, so the statement is false. The condition provided in the text relates to the relationship between the radii, indicating the ratio requires 2 times the first radius squared equals the second radius squared.

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