The two diagonals of a rhombus are of lengths 55 cm and 48 cm. If p is the perpendicular height of the rhombus, then which one of the following is correct ?

Options

A. 36 cm `lt` p `lt` 37 cm
B. 35 cm `lt` p `lt` 36 cm
C. 34 cm `lt` p `lt` 35 cm
D. 33 cm `lt` p `lt` 34 cm

Answer

To solve the problem, we need to find the perpendicular height \( P \) of the rhombus given the lengths of its diagonals. The diagonals of a rhombus bisect each other at right angles. Let's go through the solution step by step.

### Step 1: Identify the lengths of the diagonals
The lengths of the diagonals are given as:
- Diagonal \( AC = 55 \, \text{cm} \)
- Diagonal \( BD = 48 \, \text{cm} \)

### Step 2: Calculate half the lengths of the diagonals
Since the diagonals bisect each other, we can find the lengths of half the diagonals:
- \( AO = \frac{AC}{2} = \frac{55}{2} = 27.5 \, \text{cm} \)
- \( BO = \frac{BD}{2} = \frac{48}{2} = 24 \, \text{cm} \)

### Step 3: Use the Pythagorean theorem to find the side length of the rhombus
In triangle \( AOB \), we can use the Pythagorean theorem to find the length of side \( AB \):
\[ AB^2 = AO^2 + BO^2 \]
Substituting the values:
\[ AB^2 = (27.5)^2 + (24)^2 \]
Calculating each term:
\[ (27.5)^2 = 756.25 \]
\[ (24)^2 = 576 \]
Now, add these values:
\[ AB^2 = 756.25 + 576 = 1332.25 \]
Taking the square root to find \( AB \):
\[ AB = \sqrt{1332.25} \approx 36.5 \, \text{cm} \]

### Step 4: Calculate the area of the rhombus
The area \( A \) of a rhombus can also be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Where \( d_1 \) and \( d_2 \) are the lengths of the diagonals:
\[ A = \frac{1}{2} \times 55 \times 48 = 1320 \, \text{cm}^2 \]

### Step 5: Relate the area to the height
The area of the rhombus can also be expressed in terms of the base and height:
\[ A = \text{base} \times \text{height} = AB \times P \]
Substituting the known values:
\[ 1320 = 36.5 \times P \]

### Step 6: Solve for the height \( P \)
Now, we can solve for \( P \):
\[ P = \frac{1320}{36.5} \approx 36.16 \, \text{cm} \]

### Conclusion
Thus, the perpendicular height \( P \) of the rhombus is approximately \( 36.16 \, \text{cm} \).

---