Find the area of quad.ABCD in which `AB=42 , BC=21 cm, CD=29 cm ,DA =34 and `diag ,BD = 20 cm

Answer

To find the area of quadrilateral ABCD, we can divide it into two triangles, ABD and BCD, and then sum their areas. Here's a step-by-step solution:

### Step 1: Identify the sides and diagonal
Given:
- \( AB = 42 \, \text{cm} \)
- \( BC = 21 \, \text{cm} \)
- \( CD = 29 \, \text{cm} \)
- \( DA = 34 \, \text{cm} \)
- \( BD = 20 \, \text{cm} \)

### Step 2: Calculate the area of triangle ABD
1. **Calculate the semi-perimeter (s) of triangle ABD:**
\[ s = \frac{AB + AD + BD}{2} = \frac{42 + 34 + 20}{2} = \frac{96}{2} = 48 \, \text{cm} \]

2. **Use Heron's formula to find the area (A) of triangle ABD:**
\[ A_{ABD} = \sqrt{s(s - AB)(s - AD)(s - BD)} \]
Substituting the values:
\[ A_{ABD} = \sqrt{48(48 - 42)(48 - 34)(48 - 20)} = \sqrt{48 \times 6 \times 14 \times 28} \]

3. **Calculate the area:**
\[ A_{ABD} = \sqrt{48 \times 6 \times 14 \times 28} = \sqrt{36 \times 1008} = \sqrt{36288} \approx 190.5 \, \text{cm}^2 \]

### Step 3: Calculate the area of triangle BCD
1. **Calculate the semi-perimeter (s) of triangle BCD:**
\[ s = \frac{BC + CD + BD}{2} = \frac{21 + 29 + 20}{2} = \frac{70}{2} = 35 \, \text{cm} \]

2. **Use Heron's formula to find the area (A) of triangle BCD:**
\[ A_{BCD} = \sqrt{s(s - BC)(s - CD)(s - BD)} \]
Substituting the values:
\[ A_{BCD} = \sqrt{35(35 - 21)(35 - 29)(35 - 20)} = \sqrt{35 \times 14 \times 6 \times 15} \]

3. **Calculate the area:**
\[ A_{BCD} = \sqrt{35 \times 14 \times 6 \times 15} = \sqrt{44100} = 210 \, \text{cm}^2 \]

### Step 4: Calculate the total area of quadrilateral ABCD
\[ \text{Area of quadrilateral ABCD} = A_{ABD} + A_{BCD} = 190.5 + 210 = 400.5 \, \text{cm}^2 \]

### Final Answer
The area of quadrilateral ABCD is approximately \( 400.5 \, \text{cm}^2 \).

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