Find the area of a quadrilateral ABCD in which AB=3 cm, BC=4 cm, CD=4 cm, DA=5 cm and AC=5 cm

Answer

To find the area of quadrilateral ABCD, we can divide it into two triangles: triangle ABC and triangle ADC. We'll use Heron's formula to calculate the area of both triangles.

### Step 1: Calculate the area of triangle ABC
1. **Identify the sides of triangle ABC:**
- AB = 3 cm
- BC = 4 cm
- AC = 5 cm

2. **Calculate the semi-perimeter (s) of triangle ABC:**
\[ s = \frac{AB + BC + AC}{2} = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \text{ cm} \]

3. **Apply Heron's formula to find the area (A) of triangle ABC:**
\[ A = \sqrt{s(s - AB)(s - BC)(s - AC)} \]
\[ A = \sqrt{6(6 - 3)(6 - 4)(6 - 5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6 \text{ cm}^2 \]

### Step 2: Calculate the area of triangle ADC
1. **Identify the sides of triangle ADC:**
- AD = 5 cm
- DC = 4 cm
- AC = 5 cm

2. **Calculate the semi-perimeter (s) of triangle ADC:**
\[ s = \frac{AD + DC + AC}{2} = \frac{5 + 4 + 5}{2} = \frac{14}{2} = 7 \text{ cm} \]

3. **Apply Heron's formula to find the area (A) of triangle ADC:**
\[ A = \sqrt{s(s - AD)(s - DC)(s - AC)} \]
\[ A = \sqrt{7(7 - 5)(7 - 4)(7 - 5)} = \sqrt{7 \times 2 \times 3 \times 2} = \sqrt{84} = 2\sqrt{21} \approx 9.16 \text{ cm}^2 \]

### Step 3: Calculate the total area of quadrilateral ABCD
1. **Add the areas of triangles ABC and ADC:**
\[ \text{Total Area} = \text{Area of } ABC + \text{Area of } ADC = 6 \text{ cm}^2 + 9.16 \text{ cm}^2 = 15.16 \text{ cm}^2 \]

### Final Answer:
The area of quadrilateral ABCD is approximately **15.16 cm²**.

---