The area of a certain equilateral triangle is given by the expression \(\frac{\sqrt{3}}{4}\left(\frac{\mathrm{k}}{3}\right)^2\), where k is a positive...
GMAT Advanced Math : (Adv_Math) Questions
The area of a certain equilateral triangle is given by the expression \(\frac{\sqrt{3}}{4}\left(\frac{\mathrm{k}}{3}\right)^2\), where k is a positive constant. Which of the following gives the perimeter of the triangle?
\(\frac{\mathrm{k}}{3}\)
\(\mathrm{k}\)
\(\mathrm{k}\sqrt{3}\)
\(3\mathrm{k}\)
1. TRANSLATE the problem information
- Given information:
- Area of equilateral triangle = \(\frac{\sqrt{3}}{4}(\frac{\mathrm{k}}{3})^2\)
- k is a positive constant
- What we need to find: The perimeter
2. INFER the solution approach
- Key insight: We need to connect the given area expression to the triangle's side length
- Strategy: Use the standard area formula for equilateral triangles and set it equal to the given expression
- This will let us solve for the side length, then find perimeter
3. TRANSLATE the standard formula
- For an equilateral triangle with side length s: \(\mathrm{A} = \frac{\mathrm{s}^2\sqrt{3}}{4}\)
4. SIMPLIFY by setting up the equation
- Set the formulas equal: \(\frac{\mathrm{s}^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(\frac{\mathrm{k}}{3})^2\)
- Divide both sides by \(\frac{\sqrt{3}}{4}\): \(\mathrm{s}^2 = (\frac{\mathrm{k}}{3})^2\)
- Take the square root: \(\mathrm{s} = \frac{\mathrm{k}}{3}\) (positive value since side length must be positive)
5. INFER the final calculation
- Perimeter of equilateral triangle = \(3\mathrm{s}\)
- \(\mathrm{P} = 3(\frac{\mathrm{k}}{3}) = \mathrm{k}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize they need to use the standard area formula to work backwards to the side length. Instead, they might try to manipulate the given expression directly or guess based on the answer choices. This leads to confusion and random guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students set up the equation correctly but make algebraic errors, such as:
- Forgetting to divide both sides by \(\frac{\sqrt{3}}{4}\)
- Making errors when taking the square root
- Not simplifying \((\frac{\mathrm{k}}{3})^2\) correctly
This may lead them to select Choice A \((\frac{\mathrm{k}}{3})\) if they confuse the side length with the perimeter.
The Bottom Line:
This problem tests whether students can work backwards from an area expression to find geometric properties. The key breakthrough is recognizing that the given expression must equal the standard area formula, allowing you to solve for the side length systematically.
\(\frac{\mathrm{k}}{3}\)
\(\mathrm{k}\)
\(\mathrm{k}\sqrt{3}\)
\(3\mathrm{k}\)