ABCD is a parallelogram which `AB=7 cm, BC=9 cm and AC =8cm.` What is the length (in cm) of other diagonal ?

Options

A. 14
B. `14sqrt2`
C. `7`
D. `7sqrt2`

Answer

To find the length of the other diagonal \( BD \) in the parallelogram \( ABCD \) where \( AB = 7 \, \text{cm} \), \( BC = 9 \, \text{cm} \), and \( AC = 8 \, \text{cm} \), we can use the formula for the diagonals of a parallelogram.

### Step-by-step Solution:

1. **Identify the sides and diagonal of the triangle**:
We have triangle \( ABC \) with sides \( AB = 7 \, \text{cm} \), \( BC = 9 \, \text{cm} \), and diagonal \( AC = 8 \, \text{cm} \).

2. **Calculate the semi-perimeter \( s \)**:
The semi-perimeter \( s \) of triangle \( ABC \) is calculated as follows:
\[ s = \frac{AB + BC + AC}{2} = \frac{7 + 9 + 8}{2} = \frac{24}{2} = 12 \, \text{cm} \]

3. **Apply Heron's formula to find the area of triangle \( ABC \)**:
The area \( A \) of triangle \( ABC \) can be calculated using Heron's formula:
\[ A = \sqrt{s(s - AB)(s - BC)(s - AC)} \]
Substituting the values:
\[ A = \sqrt{12(12 - 7)(12 - 9)(12 - 8)} = \sqrt{12 \times 5 \times 3 \times 4} \]
\[ A = \sqrt{720} = 12\sqrt{5} \, \text{cm}^2 \]

4. **Calculate the area of parallelogram \( ABCD \)**:
The area of parallelogram \( ABCD \) is twice the area of triangle \( ABC \):
\[ \text{Area of } ABCD = 2A = 2 \times 12\sqrt{5} = 24\sqrt{5} \, \text{cm}^2 \]

5. **Use the area to find the length of diagonal \( BD \)**:
The area of the parallelogram can also be expressed in terms of the diagonals:
\[ \text{Area} = \frac{1}{2} \times AC \times BD \]
Rearranging gives:
\[ 24\sqrt{5} = \frac{1}{2} \times 8 \times BD \]
\[ 24\sqrt{5} = 4 \times BD \]
\[ BD = \frac{24\sqrt{5}}{4} = 6\sqrt{5} \, \text{cm} \]

### Final Answer:
The length of the other diagonal \( BD \) is \( 6\sqrt{5} \, \text{cm} \).