A circle is inscribed in an equilateral triangle. The circle’s area is 154 sq. cm (use π = 22/7). Find the perimeter of the equilateral triangle.

Introduction / Context:
As with similar problems, we connect the area of the inscribed circle to the triangle’s inradius r, then tie r to the triangle’s side a. From there we compute the perimeter 3a. The numbers are chosen so r comes out integral, making a and perimeter neat radicals.

Given Data / Assumptions:

• Circle area A_circ = 154 sq. cm.
• π = 22/7.
• Inradius of equilateral triangle: r = a√3 / 6.
• Perimeter P = 3a.

Concept / Approach:
Find r from πr^2 = 154; then a = 6r/√3; finally P = 3a. Keep results exact until the end to avoid rounding.

Step-by-Step Solution:
r^2 = 154 * (7/22) = 49 ⇒ r = 7r = a√3/6 ⇒ a = 6r/√3 = 42/√3 = 14√3Perimeter P = 3a = 3 * 14√3 = 42√3 cm

Verification / Alternative check:
Back-calc area: r = 7 ⇒ πr^2 = (22/7)*49 = 154 sq. cm, confirming the setup.

Why Other Options Are Wrong:
21 cm and 42 cm ignore the √3 factor; 21√3 cm is one side, not the perimeter.

Common Pitfalls:
Confusing inradius with altitude or using area of triangle instead of the circle to start; dropping √3 during simplification.

Final Answer:
42√3 cm