The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?
Question
The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?
Solution
No, the statement is not true. Here's why:
The area of a sector of a circle is given by the formula (θ/2π) * πr², where θ is the central angle of the sector and r is the radius of the circle.
If two sectors from two different circles have equal corresponding arc lengths, it means they have the same θ. However, if the radii of the two circles are different, the areas of the sectors will also be different because the radius is squared in the formula.
Therefore, even if the arc lengths are equal, the areas of the sectors can still be different if the radii of the circles are different.
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