The sides of a quadrilateral ABCD taken in order are 6 cm, 8 cm, 12 cm and 14 cm respectively and the angle between the first two sides is a right angle. Find its area. (Given, `sqrt(6) = 2.45`.)

Options

A. 82.8 `cm^(2)`
B. 84.8 `cm^(2)`
C. 81.8 `cm^(2)`
D. 86.8 `cm^(2)`

Answer

To find the area of quadrilateral ABCD with sides 6 cm, 8 cm, 12 cm, and 14 cm, where the angle between the first two sides is a right angle, we can break the quadrilateral into two triangles: triangle ABC and triangle ACD.

### Step-by-Step Solution:

1. **Identify the Triangles**:
- Triangle ABC is formed by sides AB = 6 cm, BC = 8 cm, and angle ABC = 90°.
- Triangle ACD is formed by sides AC = 10 cm (calculated later), AD = 12 cm, and CD = 14 cm.

2. **Calculate the Length of AC**:
- Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of AC.
- \( AC^2 = AB^2 + BC^2 \)
- \( AC^2 = 6^2 + 8^2 = 36 + 64 = 100 \)
- \( AC = \sqrt{100} = 10 \) cm.

3. **Calculate the Area of Triangle ABC**:
- The area of a right triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, base = 8 cm and height = 6 cm.
\[ \text{Area}_{ABC} = \frac{1}{2} \times 8 \times 6 = \frac{48}{2} = 24 \text{ cm}^2. \]

4. **Calculate the Semi-perimeter of Triangle ACD**:
- The semi-perimeter \( s \) is calculated as:
\[ s = \frac{AC + AD + CD}{2} = \frac{10 + 12 + 14}{2} = \frac{36}{2} = 18 \text{ cm}. \]

5. **Calculate the Area of Triangle ACD using Heron's Formula**:
- Heron's formula states:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
- Where \( a = 10 \) cm, \( b = 12 \) cm, and \( c = 14 \) cm.
\[ \text{Area}_{ACD} = \sqrt{18(18-10)(18-12)(18-14)} = \sqrt{18 \times 8 \times 6 \times 4}. \]
- Simplifying:
\[ = \sqrt{18 \times 192} = \sqrt{3456}. \]
- Breaking it down:
\[ = \sqrt{576 \times 6} = 24\sqrt{6}. \]
- Given \( \sqrt{6} = 2.45 \):
\[ \text{Area}_{ACD} = 24 \times 2.45 = 58.8 \text{ cm}^2. \]

6. **Calculate the Total Area of Quadrilateral ABCD**:
- The total area of quadrilateral ABCD is the sum of the areas of triangles ABC and ACD:
\[ \text{Area}_{ABCD} = \text{Area}_{ABC} + \text{Area}_{ACD} = 24 + 58.8 = 82.8 \text{ cm}^2. \]

### Final Answer:
The area of quadrilateral ABCD is **82.8 cm²**.