A trapezoid has two parallel sides measuring 25 centimeters and 41 centimeters, and a height of 16 centimeters. What is...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A trapezoid has two parallel sides measuring 25 centimeters and 41 centimeters, and a height of 16 centimeters. What is the area, in square centimeters, of the trapezoid?
1. TRANSLATE the problem information
- Given information:
- Two parallel sides: 25 cm and 41 cm
- Height: 16 cm
- Need to find: area in square centimeters
- What this tells us: We have a trapezoid with bases \(\mathrm{b_1 = 25}\), \(\mathrm{b_2 = 41}\), and height \(\mathrm{h = 16}\)
2. INFER the approach
- Since we need the area of a trapezoid and have both parallel sides plus height, we can use the trapezoid area formula
- Formula needed: \(\mathrm{A = \frac{1}{2}(b_1 + b_2)h}\)
3. SIMPLIFY the calculation
- Substitute values: \(\mathrm{A = \frac{1}{2}(25 + 41)(16)}\)
- Add the bases first: \(\mathrm{25 + 41 = 66}\)
- So: \(\mathrm{A = \frac{1}{2}(66)(16)}\)
- Divide by 2: \(\mathrm{A = 33 \times 16}\)
- Final multiplication: \(\mathrm{A = 528}\)
Answer: 528 square centimeters
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which formula to use or mix up the variables in the trapezoid formula.
Some students might try to use the triangle area formula \(\mathrm{A = \frac{1}{2}bh}\) and only use one of the parallel sides, getting \(\mathrm{A = \frac{1}{2}(25)(16) = 200}\) or \(\mathrm{A = \frac{1}{2}(41)(16) = 328}\). This leads to confusion and guessing since these aren't typical answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors in the multi-step calculation.
A common mistake is incorrectly handling the order of operations, such as calculating \(\mathrm{\frac{1}{2}(25) + (41)(16)}\) instead of \(\mathrm{\frac{1}{2}(25 + 41)(16)}\), leading to \(\mathrm{12.5 + 656 = 668.5}\), which would cause them to get stuck and randomly select an answer.
The Bottom Line:
This problem tests whether students can correctly identify the trapezoid area formula and systematically work through a multi-step calculation without arithmetic errors. The key is recognizing that both parallel sides must be added together before applying the rest of the formula.