Is the area of the largest circle that can be drawn inside a rectangle of length \(a\) cm and breadth \(b\) cm (\(a>b\)) equal to \(\pi b^2\) cm²? Why?
Step 1: We are given a rectangle of length \(a\) cm and breadth \(b\) cm, where \(a > b\). This means the rectangle is longer than it is wide.
Step 2: We want to fit the largest possible circle inside this rectangle. A circle that fits inside a rectangle is called an inscribed circle.
Step 3: The circle cannot be larger than the smaller side of the rectangle, because it has to fit completely inside. The smaller side here is the breadth \(b\).
Step 4: So, the diameter of the largest circle will be exactly equal to the breadth \(b\). That means:
\[ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, D = b \, (\text{in cm}) \]
Step 5: The radius of a circle is half the diameter. Therefore:
\[ r = \dfrac{b}{2} \, (\text{in cm}) \]
Step 6: The formula for the area of a circle is:
\[ A = \pi r^2 \]
Step 7: Substituting \(r = \dfrac{b}{2}\):
\[ A = \pi \left( \dfrac{b}{2} \right)^2 \]
Step 8: Simplify the expression:
\[ A = \pi \cdot \dfrac{b^2}{4} = \dfrac{\pi b^2}{4} \, (\text{in cm}^2) \]
Step 9: The question says the area is \(\pi b^2\) cm². But our calculation shows it is actually \(\dfrac{\pi b^2}{4}\) cm².
Conclusion: Since \(\dfrac{\pi b^2}{4} \neq \pi b^2\), the statement is False.