The area of rhombus is 169 `cm^2`. If it's one diagonal is double the other. What are the length of these diagonals?

Options

A. 39 cm, `19 cdot 5` cm
B. 26 cm, 13 cm
C. 36 cm, 18 cm
D. 40 cm, 20 cm

Answer

To find the lengths of the diagonals of the rhombus given that the area is 169 cm² and one diagonal is double the other, we can follow these steps:

### Step 1: Understand the formula for the area of a 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.

### Step 2: Set up the relationship between the diagonals
Let \( d_1 \) be the length of the shorter diagonal. According to the problem, the other diagonal \( d_2 \) is double the length of \( d_1 \). Therefore, we can express this as:
\[ d_2 = 2d_1 \]

### Step 3: Substitute the values into the area formula
Substituting \( d_2 \) into the area formula gives us:
\[ 169 = \frac{1}{2} \times d_1 \times (2d_1) \]
This simplifies to:
\[ 169 = d_1^2 \]

### Step 4: Solve for \( d_1 \)
To find \( d_1 \), we take the square root of both sides:
\[ d_1 = \sqrt{169} = 13 \text{ cm} \]

### Step 5: Find \( d_2 \)
Now that we have \( d_1 \), we can find \( d_2 \):
\[ d_2 = 2d_1 = 2 \times 13 = 26 \text{ cm} \]

### Conclusion
The lengths of the diagonals are:
- \( d_1 = 13 \text{ cm} \)
- \( d_2 = 26 \text{ cm} \)