The area of a circle inscribed in an equilateral triangle is $154 \text{ cm}^2$. Find the perimeter of the triangle.

### Concept & Reverse Engineering Perimeter When given the area of an inscribed circle, we work backward to find the radius of the circle, then use the inradius relationship to find the side of the equilateral triangle, and finally calculate its perimeter. $$ \text{Area} = \pi r^2 $$ $$ r = \frac{a}{2\sqrt{3}} $$ $$ \text{Perimeter} = 3a $$ ### Step-by-Step Solution * Given: Area of the inscribed circle $= 154 \text{ cm}^2$. * Step 1: Find the inradius $r$. $\pi r^2 = 154 \implies \frac{22}{7} r^2 = 154 \implies r^2 = 154 \times \frac{7}{22} \implies r^2 = 7 \times 7 \implies r = 7$ cm. * Step 2: Find the side $a$ of the equilateral triangle. We know $r = \frac{a}{2\sqrt{3}}$, so $7 = \frac{a}{2\sqrt{3}}$. $a = 14\sqrt{3}$ cm. * Step 3: Find the perimeter. $\text{Perimeter} = 3a = 3 \times 14\sqrt{3} = 42\sqrt{3}$ cm. * Step 4: Convert to a decimal value (using $\sqrt{3} \approx 1.732$). $\text{Perimeter} \approx 42 \times 1.732 = 72.744$ cm. * Rounding to one decimal place gives 72.7 cm. ### Exam Strategy & Shortcut Memorize that a circle with area 154 always has a radius of 7. It's one of the most common standard measurements in geometry problems. Once $r=7$ is known, $a = 14\sqrt{3}$, and Perimeter $= 42\sqrt{3}$. You can quickly estimate $42 \times 1.73 = 72.66$, pointing safely to 72.7. ### Common Pitfall Forgetting to multiply the side by 3 to get the perimeter, or prematurely rounding $\sqrt{3}$ to $1.7$, which would give $42 \times 1.7 = 71.4$, an incorrect distractor value close to Option (A). Use at least three decimal places (1.732) for accuracy. ### Final Answer Therefore, the correct answer is **72.7 cm**.