Directions : Each of the questions below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What is the area of the equilateral triangle ? I. Length of the perpendicular from one of the vertices to the opposite side is $x$ cm. II. Length of each side of the triangle is $y$ cm. III. Perimeter of the triangle is $z$ cm.

### Concept & Geometry An equilateral triangle is a completely regular polygon, meaning all its sides are equal and all its angles are $60^\circ$. Because of this strict proportionality, providing any single one-dimensional measurement (such as side length, altitude, or perimeter) uniquely defines the entire triangle, allowing you to calculate its area using standard formulas like: $$ \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 $$ ### Step-by-Step Solution Let's evaluate each statement independently: - **Statement I:** Provides the altitude (perpendicular height) as $x$. The altitude $h$ is related to the side $s$ by $h = \frac{\sqrt{3}}{2}s$. From this, we can mathematically determine the side length and then the area. This statement alone is sufficient. - **Statement II:** Provides the side length directly as $y$. We can plug this directly into the area formula. This statement alone is sufficient. - **Statement III:** Provides the perimeter as $z$. Since perimeter is $3s$, we can simply divide by $3$ to get the side length, and then find the area. This statement alone is sufficient. Since every statement works independently to yield the side length, we can answer the question using any single one of them. ### Exam Strategy & Shortcut For purely regular 2D shapes (circles, squares, equilateral triangles), any single linear dimension dictates the shape entirely. Recognize this rule to immediately select the "Any one" option without performing any calculations or writing out equations. ### Common Pitfall Students might overthink the variables $x, y,$ and $z$, assuming they are unknown variables that need to be explicitly solved for, rather than treating them as given constant values in the context of data sufficiency logic. ### Final Answer Therefore, the correct answer is **Any one of the three**.