The length of one side and the diagonal of a rectangle are 8 cm and 17 cm respectively. Find its area (in `cm^(2)`).

Options

A. 240
B. 120
C. 80
D. 160

Answer

To find the area of the rectangle given one side and the diagonal, we can follow these steps:

### Step 1: Identify the given values
We are given:
- One side (let's call it the breadth) = 8 cm
- Diagonal (let's call it AC) = 17 cm

### Step 2: Use the Pythagorean theorem
In a rectangle, the diagonal forms a right triangle with the two sides. According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Where:
- \( AC \) is the diagonal
- \( AB \) is the length we need to find
- \( BC \) is the breadth (which is given as 8 cm)

### Step 3: Substitute the known values into the equation
Substituting the values we have:
\[ 17^2 = AB^2 + 8^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 17^2 = 289 \]
\[ 8^2 = 64 \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ 289 = AB^2 + 64 \]

### Step 6: Solve for \( AB^2 \)
Rearranging the equation to solve for \( AB^2 \):
\[ AB^2 = 289 - 64 \]
\[ AB^2 = 225 \]

### Step 7: Take the square root to find \( AB \)
Now, take the square root of both sides to find \( AB \):
\[ AB = \sqrt{225} \]
\[ AB = 15 \, \text{cm} \]

### Step 8: Calculate the area of the rectangle
Now that we have both dimensions of the rectangle (length and breadth), we can calculate the area:
\[ \text{Area} = AB \times BC \]
\[ \text{Area} = 15 \, \text{cm} \times 8 \, \text{cm} \]
\[ \text{Area} = 120 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \( 120 \, \text{cm}^2 \).

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