If the length of an arc of a circle of radius \(r\) equals that of an arc of a circle of radius \(2r\), then the angle of the sector of the first circle is double the angle of the sector of the second circle. Is this statement false? Why?
Step 1: Recall the formula for the length of an arc of a circle:
\( l = r \times \theta \)
Here:
Step 2: For the first circle (radius \(r\), angle \(\alpha\)):
Arc length = \( l_1 = r \alpha \).
Step 3: For the second circle (radius \(2r\), angle \(\beta\)):
Arc length = \( l_2 = 2r \beta \).
Step 4: It is given that these two arc lengths are equal:
\( l_1 = l_2 \Rightarrow r \alpha = 2r \beta \).
Step 5: Divide both sides by \(r\) (since \(r > 0\)):
\( \alpha = 2 \beta \).
Step 6: This shows that the angle of the sector of the first circle (\(\alpha\)) is double the angle of the sector of the second circle (\(\beta\)).
Step 7: The given statement claims exactly this relationship. So the statement is true.
Step 8: But the question is asking whether this statement is false. Since we proved it is true, the correct answer is that saying it is false is itself false.