The table gives the areas of two similar polygons, P and Q, where corresponding sides are proportional.AreaPolygon P22Polygon Q198The length...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
The table gives the areas of two similar polygons, P and Q, where corresponding sides are proportional.
| Area | |
|---|---|
| Polygon P | \(\mathrm{22}\) |
| Polygon Q | \(\mathrm{198}\) |
The length of the shortest side of polygon P is \(\mathrm{6}\). What is the length of the shortest side of polygon Q?
- \(\mathrm{2}\)
- \(\mathrm{9}\)
- \(\mathrm{18}\)
- \(\mathrm{54}\)
1. TRANSLATE the problem information
- Given information:
- Polygon P has area 22, shortest side 6
- Polygon Q has area 198, shortest side unknown
- The polygons are similar (corresponding sides proportional)
2. INFER the key relationship
- For similar polygons, if the side ratio is k, then the area ratio is \(\mathrm{k^2}\)
- This means: \(\frac{\mathrm{Area\,of\,Q}}{\mathrm{Area\,of\,P}} = (\mathrm{side\,ratio})^2\)
- We can use this to find how much longer Q's sides are compared to P's sides
3. SIMPLIFY to find the area ratio
- Area ratio = \(\frac{198}{22} = 9\)
- So polygon Q has 9 times the area of polygon P
4. INFER the side ratio from area ratio
- Since area ratio = (side ratio)², we have:
- \((\mathrm{side\,ratio})^2 = 9\)
- side ratio = \(\sqrt{9} = 3\)
- This means each side of Q is 3 times as long as the corresponding side of P
5. SIMPLIFY to find the final answer
- Shortest side of Q = shortest side of P × side ratio
- Shortest side of Q = \(6 \times 3 = 18\)
Answer: C (18)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students use the area ratio directly as the side ratio, forgetting that areas scale by the square of the side ratio.
They calculate \(\frac{198}{22} = 9\) and then multiply \(6 \times 9 = 54\), thinking that if the area is 9 times larger, then the sides are also 9 times longer.
This may lead them to select Choice D (54)
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that they need to take the square root of the area ratio, but make an error with \(\sqrt{9}\).
Some students might incorrectly calculate \(\sqrt{9} = \frac{9}{3} = 3\), while others might confuse themselves and calculate something like \(\sqrt{9} = 4.5\), leading to \(6 \times 1.5 = 9\).
This may lead them to select Choice B (9)
The Bottom Line:
The key insight that makes this problem challenging is recognizing that area scales quadratically while side length scales linearly. Students who miss this fundamental relationship between similar figures will consistently arrive at incorrect answers, usually overestimating the side length.