The diagonal of a square is equal to the side of an equilateral triangle. If the area of the square is 15√3 square centimetres, what is the area of the equilateral triangle?

Introduction / Context:
This problem connects the geometry of a square with that of an equilateral triangle through the relation between their side lengths. The diagonal of the square is given to be equal to the side of the equilateral triangle, and the area of the square is known. We need to use area formulas and the relation between diagonal and side of a square to find the area of the equilateral triangle. This type of question appears often in aptitude tests to check understanding of basic area and diagonal relations.

Given Data / Assumptions:
- Area of the square is 15√3 square centimetres. - Let side of the square be s, then diagonal of the square is s√2. - The side of the equilateral triangle is equal to the diagonal of the square, so side of triangle a = s√2. - We need to find area of the equilateral triangle.

Concept / Approach:
We first find the side s of the square from its area. For a square, area = s^2. Once s is known, we write the diagonal as s√2. This diagonal is given to be the side of an equilateral triangle. For an equilateral triangle of side a, the area formula is (√3 / 4) * a^2. Substituting a = s√2 into this formula allows us to express the triangle area directly in terms of s^2. Since s^2 is already known from the square area, we can evaluate the expression and simplify.

Step-by-Step Solution:
Step 1: Let the side of the square be s. Then area of the square is s^2. Step 2: We are given that s^2 = 15√3 square centimetres. Step 3: The diagonal of the square is s√2. This diagonal is equal to the side a of the equilateral triangle. Step 4: Therefore, a = s√2. Step 5: The area of an equilateral triangle with side a is given by Area = (√3 / 4) * a^2. Step 6: Substitute a = s√2 to get Area = (√3 / 4) * (s√2)^2. Step 7: Compute (s√2)^2 = s^2 * 2. Step 8: So Area = (√3 / 4) * 2 * s^2 = (√3 / 2) * s^2. Step 9: Substitute s^2 = 15√3 into the expression: Area = (√3 / 2) * 15√3. Step 10: Multiply √3 by √3 to get 3. So Area = (1 / 2) * 15 * 3 = (1 / 2) * 45 = 45 / 2 square centimetres.

Verification / Alternative check:
You can also check sanity by approximating numerical values. Take √3 about 1.732. Then the area of the square is about 15 * 1.732 ≈ 25.98. That suggests the side of the square is roughly √26 ≈ 5.1, and the diagonal is about 5.1 * 1.414 ≈ 7.2. The side of the equilateral triangle would be about 7.2, giving an area approximately (1.732 / 4) * 7.2^2 ≈ 0.433 * 51.84 ≈ 22.4. This is close to 45 / 2 = 22.5, confirming our analytic answer is reasonable.

Why Other Options Are Wrong:
Option 45 / √2 square centimetres comes from incorrect manipulation of √2 and √3 factors. Option 45√2 square centimetres is far too large and results from squaring incorrectly or not using the correct area formula. Option 45 square centimetres doubles the correct answer and likely arises from forgetting the factor of 1/2 in the expression (√3 / 2) * s^2.

Common Pitfalls:
Students often forget or misapply the relation between the side and the diagonal of a square, sometimes using 2s instead of s√2. Another common issue is mishandling square roots when squaring expressions like s√2 or when simplifying √3 * √3. Always write the algebraic steps clearly, especially when dealing with surds, to avoid minor but costly mistakes in competitive exams.

Final Answer:
The area of the equilateral triangle is 45 / 2 square centimetres.