The adjacent sides of a parallelogram PQRS measure 34 cm and 20 cm and the diagonal PR measures 42 cm. Find the area of the parallelogram.

Options

A. `724 cm^2`
B. `342 cm^2`
C. `562 cm^2`
D. `672 cm^2`

Answer

To find the area of the parallelogram PQRS with adjacent sides measuring 34 cm and 20 cm, and diagonal PR measuring 42 cm, we can follow these steps:

### Step 1: Identify the sides and diagonal
Let:
- Side PQ = 34 cm
- Side PS = 20 cm
- Diagonal PR = 42 cm

### Step 2: Use Heron's formula to find the area of triangle PQR
First, we need to find the area of triangle PQR using Heron's formula.

1. **Calculate the semi-perimeter (s)**:
\[ s = \frac{PQ + PS + PR}{2} = \frac{34 + 20 + 42}{2} = \frac{96}{2} = 48 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula states that the area \( A \) of a triangle with sides \( a, b, c \) is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Where \( a = PQ = 34 \) cm, \( b = PS = 20 \) cm, \( c = PR = 42 \) cm.

2. **Substituting the values into the formula**:
\[ A = \sqrt{48(48-34)(48-20)(48-42)} \]
\[ A = \sqrt{48 \times 14 \times 28 \times 6} \]

### Step 4: Simplify the expression
Now, we calculate the values inside the square root:
- \( 48 - 34 = 14 \)
- \( 48 - 20 = 28 \)
- \( 48 - 42 = 6 \)

So, we have:
\[ A = \sqrt{48 \times 14 \times 28 \times 6} \]

### Step 5: Factor the numbers
To simplify \( 48 \times 14 \times 28 \times 6 \):
- \( 48 = 16 \times 3 \)
- \( 14 = 2 \times 7 \)
- \( 28 = 4 \times 7 \)
- \( 6 = 2 \times 3 \)

Combining these:
\[ 48 \times 14 \times 28 \times 6 = (16 \times 3) \times (2 \times 7) \times (4 \times 7) \times (2 \times 3) \]
\[ = 16 \times 4 \times 2^2 \times 3^2 \times 7^2 \]

### Step 6: Calculate the area of triangle PQR
Taking the square root:
\[ A = \sqrt{(16 \times 4 \times 2^2 \times 3^2 \times 7^2)} = 48 \times 7 = 336 \text{ cm}^2 \]

### Step 7: Find the area of the parallelogram
Since the area of the parallelogram is twice the area of triangle PQR:
\[ \text{Area of parallelogram} = 2 \times 336 = 672 \text{ cm}^2 \]

### Final Answer
The area of the parallelogram PQRS is **672 cm²**.

---