The area of an equilateral triangle is `6sqrt(3)` times the area of a rhombus whose one side measures 13 cm and one diagonal is 10 cm. The length of side of the triangle, in cm, is:

Options

A. `36sqrt(6)`
B. `24sqrt(6)`
C. `24sqrt(5)`
D. `12sqrt(6)`

Answer

To solve the problem step by step, we will first find the area of the rhombus and then use the relationship given in the question to find the side length of the equilateral triangle.

### Step 1: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

Given:
- One side of the rhombus = 13 cm
- One diagonal \( d_1 = 10 \) cm

We need to find the other diagonal \( d_2 \).

### Step 2: Use the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. Let \( O \) be the intersection point of the diagonals. Then we have:
- \( OA = \frac{d_1}{2} = \frac{10}{2} = 5 \) cm
- Let \( OB = x \) cm (half of the second diagonal \( d_2 \))

Using the Pythagorean theorem in triangle \( OAB \):
\[ AB^2 = OA^2 + OB^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + x^2 \]
\[ 169 = 25 + x^2 \]
\[ x^2 = 169 - 25 = 144 \]
\[ x = \sqrt{144} = 12 \text{ cm} \]
Thus, the full length of the second diagonal \( d_2 \) is:
\[ d_2 = 2x = 2 \times 12 = 24 \text{ cm} \]

### Step 3: Calculate the area of the rhombus
Now we can calculate the area of the rhombus:
\[ A = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^2 \]

### Step 4: Set up the equation for the equilateral triangle
According to the problem, the area of the equilateral triangle is \( 6\sqrt{3} \) times the area of the rhombus:
\[ \text{Area of triangle} = 6\sqrt{3} \times 120 \]
\[ \text{Area of triangle} = 720\sqrt{3} \text{ cm}^2 \]

### Step 5: Use the area formula for the equilateral triangle
The area \( A \) of an equilateral triangle with side length \( s \) is given by:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Setting this equal to the area we found:
\[ \frac{\sqrt{3}}{4} s^2 = 720\sqrt{3} \]

### Step 6: Solve for \( s^2 \)
Dividing both sides by \( \sqrt{3} \):
\[ \frac{1}{4} s^2 = 720 \]
Multiplying both sides by 4:
\[ s^2 = 2880 \]

### Step 7: Solve for \( s \)
Taking the square root of both sides:
\[ s = \sqrt{2880} \]
We can simplify \( \sqrt{2880} \):
\[ s = \sqrt{144 \times 20} = 12\sqrt{20} = 12 \times 2\sqrt{5} = 24\sqrt{5} \]

### Final Answer
The length of the side of the triangle is:
\[ s = 24\sqrt{5} \text{ cm} \]