A parallelogram has sides 15 cm and 7 cm long. The length of one of the diagonals is 20 cm. The area of the parallelogram is

Options

A. `42 cm^(2)`
B. `60 cm^(2)`
C. `84 cm^(2)`
D. `96 cm^(2)`

Answer

To find the area of the parallelogram, we can use the formula for the area of a triangle and then double it since the parallelogram can be divided into two equal triangles by one of its diagonals.

### Step-by-Step Solution:

1. **Identify the sides and diagonal of the parallelogram:**
- Let the lengths of the sides be \( a = 15 \) cm and \( b = 7 \) cm.
- The length of the diagonal \( c = 20 \) cm.

2. **Calculate the semi-perimeter (s) of the triangle formed by the sides and the diagonal:**
\[ s = \frac{a + b + c}{2} = \frac{15 + 7 + 20}{2} = \frac{42}{2} = 21 \text{ cm} \]

3. **Use Heron's formula to find the area of one triangle:**
- Heron's formula states that the area \( A \) of a triangle with sides \( a, b, c \) is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
- Substituting the values:
\[ A = \sqrt{21(21-15)(21-7)(21-20)} \]
\[ A = \sqrt{21 \times 6 \times 14 \times 1} \]
\[ A = \sqrt{21 \times 84} \]
\[ A = \sqrt{1764} = 42 \text{ cm}^2 \]

4. **Calculate the area of the parallelogram:**
- Since the area of the parallelogram is double the area of one triangle:
\[ \text{Area of the parallelogram} = 2 \times A = 2 \times 42 = 84 \text{ cm}^2 \]

### Final Answer:
The area of the parallelogram is \( 84 \text{ cm}^2 \).

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