Introduction / Context:
This problem tests understanding of the relationship between the diagonal and the area of a square. Ratio questions involving geometric figures often focus on how linear dimensions such as side or diagonal affect corresponding measures like area or volume. Here, you are given the ratio of the diagonals of two squares and asked to find the ratio of their areas, which is a classic geometry and ratio concept useful for many competitive exams.
Given Data / Assumptions:
- Two squares are considered.
- The ratio of their diagonals is 5 : 2.
- We assume both are standard squares with all interior angles equal and sides equal.
Concept / Approach:
For a square, if the side length is s, the diagonal d is given by d = s * sqrt(2). Therefore, the side length is directly proportional to the diagonal. Since the area of a square is s^2, and s is proportional to the diagonal, the area becomes proportional to the square of the diagonal. So if the ratio of diagonals is d1 : d2, then the ratio of areas will be (d1^2) : (d2^2).
Step-by-Step Solution:
1) Let the diagonals of the two squares be d1 and d2 with d1 : d2 = 5 : 2.
2) Since area of a square is proportional to the square of its diagonal, area ratio = d1^2 : d2^2.
3) Compute the squares of the ratio terms: 5^2 = 25 and 2^2 = 4.
4) Therefore, the ratio of areas of the two squares is 25 : 4.
Verification / Alternative check:
We can assume convenient values to verify. Let d2 = 2 units, then d1 = 5 units. For a square, side s = d / sqrt(2). So s1 = 5 / sqrt(2) and s2 = 2 / sqrt(2). Then area1 = (5 / sqrt(2))^2 = 25 / 2, and area2 = (2 / sqrt(2))^2 = 4 / 2 = 2. The ratio area1 : area2 = (25 / 2) : 2 = 25 : 4. This confirms our earlier reasoning and shows that choosing actual numerical values is a valid alternative method.
Why Other Options Are Wrong:
Option A (5 : 6) and option C (5 : 4) treat the ratio as if area were directly proportional to the diagonal instead of the square of the diagonal. Option D (125 : 8) would correspond to cubing the ratio rather than squaring it. Option E (4 : 25) is simply the inverse of the correct ratio and would only be valid if the question asked for the ratio of the smaller area to the larger area in reversed order. Thus, only 25 : 4 correctly represents the ratio of the larger square’s area to the smaller square’s area.
Common Pitfalls:
Many learners mistakenly assume that the ratio of areas is the same as the ratio of diagonals, forgetting that area depends on the square of a linear dimension. Another common pitfall is to confuse diagonal-based formulas with side-based formulas and perform unnecessary calculations. Always remember that when a linear dimension changes in ratio a : b, the area changes in the ratio a^2 : b^2 for similar figures like squares.
Final Answer:
The ratio of the areas of the two squares is
25 : 4.
Discussion & Comments