Triangle XYZ is similar to triangle PQR. If the ratio of corresponding sides XY : PQ is 5 : 1 and the area of triangle PQR is 5 square cm, what is the area (in square cm) of triangle XYZ?

Introduction / Context:
This geometry question focuses on the relationship between the side lengths and the areas of similar triangles. When two triangles are similar, the ratio of their areas is the square of the ratio of any pair of corresponding sides. This principle appears frequently in aptitude problems and helps solve area questions quickly.

Given Data / Assumptions:

• Triangle XYZ is similar to triangle PQR.
• The ratio of corresponding sides XY : PQ is 5 : 1.
• The area of triangle PQR is 5 square cm.
• We need to find the area of triangle XYZ.
• Similarity implies all corresponding side ratios are 5 : 1.

Concept / Approach:
For two similar triangles, if the ratio of corresponding side lengths is k, then the ratio of their areas is k^2. Here, the scale factor from triangle PQR to triangle XYZ is 5 (since XY is five times PQ). Therefore, the area of triangle XYZ is 5^2 times the area of triangle PQR. This uses the fact that area scales with the square of linear dimensions.

Step-by-Step Solution:
Step 1: Note that XY : PQ = 5 : 1. Step 2: Let k be the linear scale factor from PQR to XYZ, so k = 5. Step 3: For similar triangles, area(XYZ) / area(PQR) = k^2 = 5^2 = 25. Step 4: Given area(PQR) = 5 square cm, compute area(XYZ) = 25 * 5 = 125 square cm. Step 5: Therefore, the area of triangle XYZ is 125 square cm.

Verification / Alternative check:
Imagine scaling triangle PQR by a factor of 5 in all directions. Every side becomes five times longer, so any base or height pair used to compute the area is multiplied by 5. Since area is roughly base * height / 2, both base and height are scaled, leading to a factor of 5 * 5 = 25 in area. This intuitive picture matches the algebraic rule that area scales with the square of the scale factor and confirms the answer of 125 square cm.

Why Other Options Are Wrong:
120 and 100 square cm are close but not equal to 25 times 5; they may come from miscalculating 5^2 or the multiplication with area(PQR). 64 square cm corresponds to a scale factor of 8 between sides, which contradicts the given ratio 5 : 1. 75 square cm corresponds to an area ratio of 15, which is not a square number and therefore cannot arise from a consistent side ratio of 5 : 1 between similar triangles.

Common Pitfalls:
A common mistake is to multiply the area by the side ratio directly (using 5 instead of 5^2), which would give 25 square cm rather than 125. Another error is to misinterpret the ratio and treat it as PQ : XY rather than XY : PQ, which reverses the scaling direction. Always confirm which triangle is larger and square the correct linear scale factor when dealing with areas of similar figures.

Final Answer:
The area of triangle XYZ is 125 square cm.