Exploring Some Geometric ThemesClass 8 Mathematics Assertion & Reason Questions

Assertion (A): In the construction of the Sierpinski Carpet starting from a single square, the number of remaining squares at Step 2 is 64.

Reason (R): The number of remaining squares at step 'n' in the Sierpinski Carpet construction is given by the formula $R_n = 8^n$.

Assertion (A): The perimeter of the shape in the construction of the Koch Snowflake increases at each step.

Reason (R): At each step, every line segment is replaced by 4 new segments, each being one-third the length of the original segment. The new perimeter becomes $\frac{4}{3}$ times the previous perimeter.

Assertion (A): The total area of a Sierpinski Gasket, constructed from an initial equilateral triangle of area 1, approaches zero as the number of steps increases infinitely.

Reason (R): At each step 'n', the remaining area is given by $A_n = (\frac{3}{4})^n$, and the limit of $(\frac{3}{4})^n$ as n approaches infinity is 0.