In a rhombus, one of the diagonals is 70% of the other diagonal. What is the ratio of the area of the rhombus to the square of the length of the larger diagonal?

Introduction / Context:
This question applies the formula for the area of a rhombus in terms of its diagonals and asks for a ratio involving the square of the longer diagonal. It combines knowledge of geometry and percent relationships. The rhombus is a quadrilateral with all sides equal, and its diagonals intersect at right angles. The area can be conveniently expressed using the product of the diagonals.

Given Data / Assumptions:

• In the rhombus, let the longer diagonal be denoted by D.
• The shorter diagonal is 70% of the longer one, that is 0.7 * D.
• Area of a rhombus is (1 / 2) * (product of the diagonals).
• We need the ratio: (area of rhombus) : (D^2).

Concept / Approach:
We use the formula for the area of a rhombus: Area = (1 / 2) * d1 * d2, where d1 and d2 are the diagonals. Here, d1 = D (larger diagonal) and d2 = 0.7 * D. Substituting these into the area formula, we obtain area in terms of D^2. Then we form the ratio of this area to D^2. Since D is positive, it cancels out properly in the ratio, leaving a numerical fraction which we convert into a ratio form.

Step-by-Step Solution:
Let the longer diagonal be D. Then the shorter diagonal is 70% of D, that is 0.7 * D. Area of the rhombus = (1 / 2) * d1 * d2 = (1 / 2) * D * (0.7 * D). So area = (1 / 2) * 0.7 * D^2 = 0.35 * D^2. We need the ratio (area) : (D^2). So ratio = 0.35 * D^2 : D^2. Cancel D^2 to get 0.35 : 1. Write 0.35 as 35 / 100, which simplifies to 7 / 20. Hence the required ratio = 7 : 20.

Verification / Alternative check:
Choose a convenient value for D, for example D = 20 units. Then the shorter diagonal is 70% of 20, which is 14 units. Area = (1 / 2) * 20 * 14 = 140 square units. The square of the longer diagonal is D^2 = 400. The ratio area : D^2 is then 140 : 400. Dividing both by 20 gives 7 : 20, confirming our earlier algebraic result.

Why Other Options Are Wrong:
Ratios like 3 : 10, 3 : 20, 7 : 10 or 1 : 2 do not match the fraction 0.35 when expressed as a ratio, and therefore do not accurately represent area : D^2 in this configuration. Only 7 : 20 corresponds to 0.35 in simplest ratio form.

Common Pitfalls:
Some students mistakenly interpret 70% as 0.7 added to 1, using 1.7 * D instead of 0.7 * D for the shorter diagonal. Others plug the percentage incorrectly into the area formula or attempt to square 0.7 incorrectly. Another frequent mistake is to forget to simplify the decimal 0.35 into a clean ratio. Always convert percentages into decimals or fractions carefully and follow the formula for area step by step.

Final Answer:
The ratio of the area of the rhombus to the square of the larger diagonal is 7 : 20.