Find the area of a parallelogram ABCD in which AB = 8 cm, BC = 15 cm and diagonal AC = 17 cm.

Answer

To find the area of the parallelogram ABCD with given dimensions, we will use Heron's formula. Here are the steps:

### Step 1: Identify the sides and diagonal
We are given:
- AB = 8 cm (side)
- BC = 15 cm (side)
- AC = 17 cm (diagonal)

### Step 2: Draw the parallelogram
Visualize the parallelogram ABCD with the given dimensions. Label the points A, B, C, and D accordingly.

### Step 3: Calculate the semi-perimeter (S) of triangle ABC
The semi-perimeter \( S \) is calculated using the formula:
\[ S = \frac{AB + BC + AC}{2} \]
Substituting the values:
\[ S = \frac{8 + 15 + 17}{2} = \frac{40}{2} = 20 \text{ cm} \]

### Step 4: Apply Heron's formula to find the area of triangle ABC
Heron's formula states that the area \( A \) of a triangle can be calculated as:
\[ A = \sqrt{S \times (S - AB) \times (S - BC) \times (S - AC)} \]
Substituting the values:
\[ A = \sqrt{20 \times (20 - 8) \times (20 - 15) \times (20 - 17)} \]
Calculating each term:
- \( S - AB = 20 - 8 = 12 \)
- \( S - BC = 20 - 15 = 5 \)
- \( S - AC = 20 - 17 = 3 \)

Now substituting these values back into the formula:
\[ A = \sqrt{20 \times 12 \times 5 \times 3} \]

### Step 5: Calculate the area of triangle ABC
Calculating the product:
\[ 20 \times 12 = 240 \]
\[ 240 \times 5 = 1200 \]
\[ 1200 \times 3 = 3600 \]
Now, taking the square root:
\[ A = \sqrt{3600} = 60 \text{ cm}^2 \]

### Step 6: Find the area of the parallelogram ABCD
Since the area of the parallelogram is twice the area of triangle ABC:
\[ \text{Area of parallelogram ABCD} = 2 \times A = 2 \times 60 = 120 \text{ cm}^2 \]

### Final Answer:
The area of the parallelogram ABCD is \( 120 \text{ cm}^2 \).

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