One diagonal of a rhombus is half the other. If the length of the side of the rhombus is 20 cm, what is the area of the rhombus ?

Options

A. 320 square cm.
B. 160 sqaure cm.
C. 240 square cm
D. 360 square cm.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the properties of the rhombus
A rhombus has two diagonals that bisect each other at right angles. We are given that one diagonal is half the length of the other.

### Step 2: Define the diagonals
Let the length of the longer diagonal \( AC \) be \( 4x \) cm. Therefore, the shorter diagonal \( BD \) will be \( 2x \) cm.

### Step 3: Use the properties of the rhombus
Since the diagonals bisect each other, we can find the lengths of half of each diagonal:
- \( OD = \frac{BD}{2} = \frac{2x}{2} = x \) cm
- \( OC = \frac{AC}{2} = \frac{4x}{2} = 2x \) cm

### Step 4: Apply the Pythagorean theorem
In triangle \( ODC \), where \( OD \) and \( OC \) are the two sides and \( CD \) is the hypotenuse (which is also the side of the rhombus), we can apply the Pythagorean theorem:
\[ OC^2 + OD^2 = CD^2 \]
Substituting the values we have:
\[ (2x)^2 + (x)^2 = 20^2 \]
This simplifies to:
\[ 4x^2 + x^2 = 400 \]
\[ 5x^2 = 400 \]

### Step 5: Solve for \( x \)
Dividing both sides by 5:
\[ x^2 = \frac{400}{5} = 80 \]
Taking the square root:
\[ x = \sqrt{80} = 4\sqrt{5} \text{ cm} \]

### Step 6: Find the lengths of the diagonals
Now we can find the lengths of the diagonals:
- \( AC = 4x = 4(4\sqrt{5}) = 16\sqrt{5} \) cm
- \( BD = 2x = 2(4\sqrt{5}) = 8\sqrt{5} \) cm

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the lengths of the diagonals:
\[ A = \frac{1}{2} \times (16\sqrt{5}) \times (8\sqrt{5}) \]
Calculating this gives:
\[ A = \frac{1}{2} \times 128 \times 5 = 320 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is \( 320 \text{ cm}^2 \).

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