Find the area of a rhombus one side of which measures 20 cm and one of whose diagonals is 24 cm.

Options

A. 384 `cm^(2)`
B. 364 `cm^(2)`
C. 354 `cm^(2)`
D. 374 `cm^(2)`

Answer

To find the area of a rhombus with one side measuring 20 cm and one diagonal measuring 24 cm, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus has the following properties:
- All sides are equal.
- The diagonals bisect each other at right angles (90 degrees).

### Step 2: Draw the rhombus and label the points
Let’s label the rhombus as ABCD, where:
- AB = BC = CD = DA = 20 cm (the sides of the rhombus)
- AC = 24 cm (one of the diagonals)
- Let O be the point where the diagonals intersect.

### Step 3: Calculate the lengths of the diagonals
Since the diagonals bisect each other, we can denote:
- AO = OC = 12 cm (since AC = 24 cm, each half is 12 cm)
- Let BD be the other diagonal, which we will denote as 2 * OB (since OB = OD).

### Step 4: Apply the Pythagorean theorem
In triangle AOB, we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + OB^2 \]
Substituting the known values:
\[ 20^2 = 12^2 + OB^2 \]
\[ 400 = 144 + OB^2 \]
\[ OB^2 = 400 - 144 \]
\[ OB^2 = 256 \]
\[ OB = \sqrt{256} = 16 \text{ cm} \]

### Step 5: Calculate the length of the other diagonal
Since OB = 16 cm, then:
\[ BD = 2 \times OB = 2 \times 16 = 32 \text{ cm} \]

### Step 6: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times (AC) \times (BD) \]
Substituting the values we have:
\[ A = \frac{1}{2} \times 24 \times 32 \]
\[ A = \frac{1}{2} \times 768 \]
\[ A = 384 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is \( 384 \text{ cm}^2 \).

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