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

Options

A. `220 cm^(2)`
B. `150 cm^(2)`
C. `120 cm^(2)`
D. `175 cm^(2)`

Answer

To find the area of parallelogram ABCD with given dimensions, we can follow these steps:

### Step 1: Identify the given dimensions
We have:
- AB = 8 cm
- BC = 15 cm
- Diagonal AC = 17 cm

### Step 2: 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 3: Use Heron's formula to find the area of triangle ABC
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a = AB \), \( b = BC \), and \( c = AC \). Substituting the values:
\[ A = \sqrt{20 \times (20 - 8) \times (20 - 15) \times (20 - 17)} \]
Calculating each term:
\[ A = \sqrt{20 \times 12 \times 5 \times 3} \]

### Step 4: Calculate the product inside the square root
Calculating the product:
\[ 20 \times 12 = 240 \]
\[ 240 \times 5 = 1200 \]
\[ 1200 \times 3 = 3600 \]
So,
\[ A = \sqrt{3600} \]
Calculating the square root:
\[ A = 60 \text{ cm}^2 \]

### Step 5: Calculate the area of parallelogram ABCD
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 parallelogram ABCD is \( 120 \text{ cm}^2 \).

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