Find the area `(" in" cm^(2))` of a rhombus whose side is 17 cm and one of its diagonals is 30 cm .

Options

A. 510
B. 600
C. 240
D. 350

Answer

To find the area of a rhombus given its side length and one of its diagonals, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has two diagonals that bisect each other at right angles. If we denote the diagonals as \(d_1\) and \(d_2\), then we know that:
- The diagonals bisect each other, meaning each diagonal is divided into two equal parts.
- The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]

### Step 2: Identify the given values
From the problem, we know:
- The side length of the rhombus \(s = 17 \, \text{cm}\)
- One of the diagonals \(d_1 = 30 \, \text{cm}\)

### Step 3: Calculate half of the given diagonal
Since the diagonal \(d_1\) is 30 cm, each half of this diagonal will be:
\[ \frac{d_1}{2} = \frac{30}{2} = 15 \, \text{cm} \]

### Step 4: Use the Pythagorean theorem to find the other diagonal
Let \(d_2\) be the other diagonal. The diagonals bisect each other at right angles, so we can form a right triangle using half of each diagonal and the side of the rhombus. In this triangle:
- One leg is \(\frac{d_1}{2} = 15 \, \text{cm}\)
- The other leg is \(\frac{d_2}{2}\)
- The hypotenuse is the side of the rhombus, which is \(17 \, \text{cm}\)

Using the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ 17^2 = 15^2 + \left(\frac{d_2}{2}\right)^2 \]
Calculating the squares:
\[ 289 = 225 + \left(\frac{d_2}{2}\right)^2 \]
Subtracting \(225\) from both sides:
\[ 289 - 225 = \left(\frac{d_2}{2}\right)^2 \]
\[ 64 = \left(\frac{d_2}{2}\right)^2 \]
Taking the square root:
\[ \frac{d_2}{2} = 8 \, \text{cm} \]
Thus, the full length of the second diagonal \(d_2\) is:
\[ d_2 = 2 \times 8 = 16 \, \text{cm} \]

### Step 5: Calculate the area of the rhombus
Now that we have both diagonals, we can calculate the area:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 30 \times 16 \]
Calculating the area:
\[ \text{Area} = \frac{1}{2} \times 480 = 240 \, \text{cm}^2 \]

### Final Answer
The area of the rhombus is \(240 \, \text{cm}^2\).

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