Find the length of a rectangle whose diagonal and area is 25 cm and 168 `cm^2` respectively :

Options

A. 31 cm
B. 24 cm
C. 17 cm
D. 9 cm

Answer

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

### Step 1: Define Variables
Let the length of the rectangle be \( L \) and the breadth be \( B \).

### Step 2: Use the Diagonal Formula
The diagonal \( D \) of a rectangle can be expressed using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]
Given that the diagonal is 25 cm, we can write:
\[ \sqrt{L^2 + B^2} = 25 \]
Squaring both sides gives us:
\[ L^2 + B^2 = 625 \quad \text{(Equation 1)} \]

### Step 3: Use the Area Formula
The area \( A \) of a rectangle is given by:
\[ A = L \times B \]
Given that the area is 168 cm², we can write:
\[ L \times B = 168 \quad \text{(Equation 2)} \]

### Step 4: Express \( B \) in Terms of \( L \)
From Equation 2, we can express \( B \) as:
\[ B = \frac{168}{L} \]

### Step 5: Substitute \( B \) in Equation 1
Substituting \( B \) in Equation 1:
\[ L^2 + \left(\frac{168}{L}\right)^2 = 625 \]
This simplifies to:
\[ L^2 + \frac{28224}{L^2} = 625 \]

### Step 6: Multiply through by \( L^2 \) to eliminate the fraction
Multiplying through by \( L^2 \) gives:
\[ L^4 - 625L^2 + 28224 = 0 \]
Let \( x = L^2 \). Then the equation becomes:
\[ x^2 - 625x + 28224 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{625 \pm \sqrt{625^2 - 4 \cdot 1 \cdot 28224}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 625^2 = 390625 \]
\[ 4 \cdot 28224 = 112896 \]
\[ 625^2 - 4 \cdot 28224 = 390625 - 112896 = 277729 \]
Now, taking the square root:
\[ \sqrt{277729} = 527 \]
Thus, we have:
\[ x = \frac{625 \pm 527}{2} \]

Calculating the two possible values:
1. \( x = \frac{1152}{2} = 576 \)
2. \( x = \frac{98}{2} = 49 \)

### Step 8: Find \( L \) from \( x \)
Since \( x = L^2 \):
1. If \( L^2 = 576 \), then \( L = 24 \) (since length must be positive).
2. If \( L^2 = 49 \), then \( L = 7 \).

### Step 9: Determine the Corresponding \( B \)
Using \( L = 24 \):
\[ B = \frac{168}{24} = 7 \]
Using \( L = 7 \):
\[ B = \frac{168}{7} = 24 \]

### Conclusion
The length of the rectangle is \( L = 24 \) cm (the other dimension is the breadth \( B = 7 \) cm).