Understanding the Rhombus and Diagonals

A rhombus is a quadrilateral where all four sides are equal in length. A key property of a rhombus is that its diagonals bisect each other at right angles. The area of a rhombus can be calculated using the lengths of its diagonals.

Let the lengths of the two diagonals of the rhombus be $d_1$ and $d_2$. The formula for the area of a rhombus is:

Area $= \frac{1}{2} \times d_1 \times d_2$

Relating the Rhombus Diagonals

The problem states that one of the diagonals is 70 percent of the other. Let's assume $d_1$ is the larger diagonal and $d_2$ is the smaller diagonal. According to the question:

$d_2 = 70\%$ of $d_1$

To work with this percentage, we convert it to a decimal or a fraction:

$70\% = \frac{70}{100} = 0.7$ or $\frac{7}{10}$

So, the relationship between the diagonals is:

$d_2 = 0.7 \times d_1$

Calculating the Area in terms of the Larger Diagonal

Now, we substitute the expression for $d_2$ into the area formula:

Area $= \frac{1}{2} \times d_1 \times d_2$

Area $= \frac{1}{2} \times d_1 \times (0.7 d_1)$

Area $= \frac{1}{2} \times 0.7 \times d_1^2$

Area $= 0.5 \times 0.7 \times d_1^2$

Area $= 0.35 \times d_1^2$

So, the area of the rhombus is $0.35$ times the square of the length of the larger diagonal.

Finding the Ratio of Area to Square of Larger Diagonal

The question asks for the ratio of the area of the rhombus to the square of the length of the larger diagonal. This can be written as:

Ratio $= \frac{\text{Area}}{(\text{Larger Diagonal})^2}$

Ratio $= \frac{0.35 \times d_1^2}{d_1^2}$

The term $d_1^2$ cancels out from the numerator and denominator (assuming $d_1 \neq 0$, which must be true for a rhombus to exist):

Ratio $= 0.35$

Expressing the Ratio as a Fraction

The ratio $0.35$ needs to be expressed as a fraction to match the options. We convert the decimal to a fraction:

$0.35 = \frac{35}{100}$

Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

Numerator: $35 \div 5 = 7$

Denominator: $100 \div 5 = 20$

So, the simplified fraction is $\frac{7}{20}$.

The ratio of the area of the rhombus to the square of the length of the larger diagonal is $7:20$.

Conclusion on Rhombus Area Ratio

Based on the calculations, the ratio of the area of the rhombus to the square of the length of the larger diagonal is $7:20$. This matches one of the given options.

Revision Table: Key Rhombus Concepts

Additional Information on Rhombus Properties

Besides the area formula using diagonals, here are some other important properties of a rhombus that might be useful in related problems:

• All sides are congruent.
• Opposite sides are parallel (a rhombus is a type of parallelogram).
• Opposite angles are equal.
• Adjacent angles are supplementary (sum up to 180 degrees).
• The diagonals bisect the angles of the rhombus.
• The perimeter of a rhombus is $4 \times \text{side length}$.
• The rhombus is formed by two congruent triangles sharing a base side, or four congruent right-angled triangles formed by the diagonals.