The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?
Step 1: Recall the formula for the area of a sector:
\( A = \tfrac{1}{2} r^2 \theta \)
Here, \(r\) is the radius (in SI unit: metre), and \(\theta\) is the angle at the centre (in radians).
Step 2: We also know that the arc length of a sector is:
\( l = r \theta \)
Step 3: Substitute \(\theta = \tfrac{l}{r}\) into the area formula:
\( A = \tfrac{1}{2} r^2 \cdot \tfrac{l}{r} \)
Step 4: Simplify the expression:
\( A = \tfrac{1}{2} r \cdot l \)
Step 5: This shows that the area of a sector depends on both the radius \(r\) (metres) and the arc length \(l\) (metres).
Step 6: If two sectors have the same arc length \(l\), but the radii of the circles are different, then their areas will be different because the formula has \(r\) multiplied with \(l\).
Step 7: Therefore, the statement that their areas are equal is False, unless the radii are also the same.