The area of a right triangular sail is 1{,260} square feet. The length of the longer leg of the triangle,...
GMAT Advanced Math : (Adv_Math) Questions
The area of a right triangular sail is \(\mathrm{1{,}260}\) square feet. The length of the longer leg of the triangle, \(\mathrm{x}\), in feet, is \(\mathrm{12}\) feet longer than the length of the shorter leg. Which equation represents this situation?
\(0 = \mathrm{x}^2 + 12\mathrm{x} - 2{,}520\)
\(0 = \mathrm{x}^2 - 12\mathrm{x} - 2{,}520\)
\(0 = \mathrm{x}^2 - 12\mathrm{x} - 1{,}260\)
\(0 = \mathrm{x}^2 + 12\mathrm{x} - 1{,}260\)
1. TRANSLATE the problem information
- Given information:
- Area of right triangle = 1,260 square feet
- Longer leg = x feet
- The longer leg is 12 feet longer than the shorter leg means: shorter leg = x - 12 feet
2. INFER the appropriate formula strategy
- For a right triangle, we can use the two legs as base and height in the area formula
- \(\mathrm{Area = \frac{1}{2} \times base \times height}\) works perfectly here since we know both legs
3. TRANSLATE the area relationship into an equation
Set up the equation:
\(\mathrm{1,260 = \frac{1}{2} \times x \times (x - 12)}\)
4. SIMPLIFY by clearing the fraction
Multiply both sides by 2:
\(\mathrm{2,520 = x(x - 12)}\)
5. SIMPLIFY by expanding and rearranging
Expand the right side:
\(\mathrm{2,520 = x^2 - 12x}\)
Rearrange to standard form:
\(\mathrm{0 = x^2 - 12x - 2,520}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor TRANSLATE reasoning: Students often confuse which leg should be represented by x, thinking the shorter leg is x and the longer leg is x + 12.
This leads them to set up: shorter leg = x, longer leg = \(\mathrm{x + 12}\), giving them the equation \(\mathrm{1,260 = \frac{1}{2} \times x \times (x + 12)}\), which expands to \(\mathrm{2,520 = x^2 + 12x}\), and rearranges to \(\mathrm{0 = x^2 + 12x - 2,520}\).
This may lead them to select Choice A (\(\mathrm{0 = x^2 + 12x - 2,520}\)).
Second Most Common Error:
Weak SIMPLIFY execution: Students correctly set up the initial equation but make algebraic errors when expanding or rearranging, such as forgetting to multiply the area by 2 when clearing the fraction.
This leads to working with \(\mathrm{1,260 = x^2 - 12x}\) instead of \(\mathrm{2,520 = x^2 - 12x}\), giving them \(\mathrm{0 = x^2 - 12x - 1,260}\).
This may lead them to select Choice C (\(\mathrm{0 = x^2 - 12x - 1,260}\)).
The Bottom Line:
The key challenge is correctly interpreting 12 feet longer than and consistently applying algebraic operations without sign errors. Success requires careful translation of the relationship and methodical algebraic manipulation.
\(0 = \mathrm{x}^2 + 12\mathrm{x} - 2{,}520\)
\(0 = \mathrm{x}^2 - 12\mathrm{x} - 2{,}520\)
\(0 = \mathrm{x}^2 - 12\mathrm{x} - 1{,}260\)
\(0 = \mathrm{x}^2 + 12\mathrm{x} - 1{,}260\)