Calculate the area of quadrilateral ABCD In which `angle A = 90^(@) , AB = 16 cm . AD = 12 cm and BC = CD = 12.5 cm `

Answer

To calculate the area of quadrilateral ABCD, we can divide it into two triangles: triangle ABD and triangle BCD.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Angle A = 90°
- AB = 16 cm
- AD = 12 cm
- BC = 12.5 cm
- CD = 12.5 cm

2. **Calculate the Area of Triangle ABD**:
- Since angle A is 90°, triangle ABD is a right triangle.
- The area of triangle ABD can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, we can take AB as the base and AD as the height:
\[ \text{Area}_{ABD} = \frac{1}{2} \times AB \times AD = \frac{1}{2} \times 16 \times 12 \]
- Calculate:
\[ \text{Area}_{ABD} = \frac{1}{2} \times 192 = 96 \text{ cm}^2 \]

3. **Calculate the Length of Diagonal BD**:
- Using the Pythagorean theorem for triangle ABD:
\[ BD = \sqrt{AD^2 + AB^2} \]
- Substitute the values:
\[ BD = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm} \]

4. **Calculate the Area of Triangle BCD**:
- We can use Heron's formula to find the area of triangle BCD.
- First, calculate the semi-perimeter (s):
\[ s = \frac{BC + CD + BD}{2} = \frac{12.5 + 12.5 + 20}{2} = 22.5 \text{ cm} \]
- Now, apply Heron's formula:
\[ \text{Area}_{BCD} = \sqrt{s(s - BC)(s - CD)(s - BD)} \]
- Substitute the values:
\[ \text{Area}_{BCD} = \sqrt{22.5 \times (22.5 - 12.5) \times (22.5 - 12.5) \times (22.5 - 20)} \]
- Calculate:
\[ = \sqrt{22.5 \times 10 \times 10 \times 2.5} = \sqrt{5625} = 75 \text{ cm}^2 \]

5. **Calculate the Total Area of Quadrilateral ABCD**:
- Now, add the areas of triangles ABD and BCD:
\[ \text{Area}_{ABCD} = \text{Area}_{ABD} + \text{Area}_{BCD} = 96 + 75 = 171 \text{ cm}^2 \]

### Final Answer:
The area of quadrilateral ABCD is **171 cm²**.