A triangle of area $9 y \text{ cm}^2$ has been drawn such that its area is equal to the area of an equilateral triangle of side $6\text{ cm}$. The value of $y$ would be

### Concept & Equilateral Triangle Area The area of an equilateral triangle can be determined if the side length is known using the standard formula: $$ \text{Area} = \frac{\sqrt{3}}{4} a^2 $$ where $a$ is the side of the triangle. ### Step-by-Step Solution 1. Identify the given side length of the equilateral triangle: $a = 6\text{ cm}$. 2. Calculate the area of this equilateral triangle: $\text{Area} = \frac{\sqrt{3}}{4} \times (6)^2$ $\text{Area} = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}\text{ cm}^2$. 3. We are given that the area of another triangle is $9 y \text{ cm}^2$ and it equals the area of this equilateral triangle. 4. Equate the two expressions for the area: $9 y = 9\sqrt{3}$ 5. Solve for $y$: $y = \sqrt{3}$. ### Exam Strategy & Shortcut Memorize the formula for the area of an equilateral triangle. When setting up the equation $9y = \frac{\sqrt{3}}{4}(36)$, the 9s immediately cancel out, leaving $y = \sqrt{3}$ instantly. ### Common Pitfall A common mistake is confusing the formula for the height of an equilateral triangle ($\frac{\sqrt{3}}{2}a$) with the formula for the area. ### Final Answer Therefore, the correct answer is **$\sqrt{3}$**.