The areas of two similar triangles are in the ratio 5 : 7. What is the ratio of the lengths of their corresponding sides?

Introduction / Context:
This question tests the relationship between areas and side lengths of similar triangles. In similar figures, the ratio of areas is the square of the ratio of corresponding sides. Recognising and applying this proportionality is a foundational skill in similarity and scaling problems in geometry.

Given Data / Assumptions:

• Two triangles are similar.
• The ratio of their areas is 5 : 7.
• We need the ratio of the lengths of their corresponding sides.
• Both triangles are non degenerate and similarity is exact.
• Standard similarity relations between side and area ratios apply.

Concept / Approach:
For two similar triangles, if the ratio of corresponding sides is k, then the ratio of their areas is k^2. Conversely, if we know the area ratio, the side ratio is the positive square root of that ratio. Here, the given area ratio is 5 : 7, so the side ratio will be √5 : √7. This captures the scaling factor that relates side lengths while preserving the fixed area ratio.

Step-by-Step Solution:
Step 1: Let the ratio of corresponding sides be a : b. Step 2: For similar triangles, area ratio = (side ratio)^2, so (Area₁ / Area₂) = (a / b)^2. Step 3: Given area ratio is 5 : 7, so (a / b)^2 = 5 / 7. Step 4: Take the positive square root: a / b = √(5 / 7) = √5 / √7. Step 5: Therefore the ratio of corresponding side lengths is √5 : √7.

Verification / Alternative check:
Assume for simplicity that side lengths of the first triangle are √5 times some base unit and side lengths of the second triangle are √7 times the same base unit. Then, since area is proportional to side squared, area of the first triangle will be proportional to (√5)^2 = 5 and area of the second to (√7)^2 = 7. This reproduces the given area ratio 5 : 7, confirming that √5 : √7 is the correct side ratio.

Why Other Options Are Wrong:
Option 5 : 7 mistakenly equates side ratio with area ratio. Option 25 : 49 corresponds to squaring the area ratio and is thus too large. The ratio 125 : 343 is 5^3 : 7^3 and would be related to some volume scaling, not area. The ratio 5√5 : 7√7 would produce an area ratio of 25 * 5 : 49 * 7 = 125 : 343, not 5 : 7. Only √5 : √7 produces the correct area ratio when squared.

Common Pitfalls:
Many learners confuse whether they should square or take square roots of the given ratio. A good rule is that areas scale with the square of linear dimensions, and volumes scale with the cube. When an area ratio is given and a length ratio is asked for, you must take the square root. Writing a small example or using a generic scale factor k in formulas helps to remember and apply this correctly.

Final Answer:
The ratio of corresponding side lengths is √5 : √7.