ABCD is a quadrilateral AC=19 cm. The lengths of perpendicular from B and D on AC are 5 cm and 7 cm respectively . Then , the area of ABCD ( in `cm^(2)`) is :

Options

A. 162
B. 144
C. 228
D. 114

Answer

To find the area of quadrilateral ABCD, we can break it down into two triangles: triangle ABC and triangle ADC. We will use the formula for the area of a triangle, which is given by:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

### Step-by-Step Solution:

1. **Identify the Base and Heights:**
- The base AC is given as 19 cm.
- The height from point B to line AC (perpendicular) is 5 cm.
- The height from point D to line AC (perpendicular) is 7 cm.

2. **Calculate the Area of Triangle ABC:**
- Using the formula for the area of a triangle:
\[ \text{Area}_{ABC} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Here, the base is AC (19 cm) and the height is from B (5 cm):
\[ \text{Area}_{ABC} = \frac{1}{2} \times 19 \times 5 = \frac{95}{2} \text{ cm}^2 \]

3. **Calculate the Area of Triangle ADC:**
- Similarly, for triangle ADC:
\[ \text{Area}_{ADC} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Here, the base is AC (19 cm) and the height is from D (7 cm):
\[ \text{Area}_{ADC} = \frac{1}{2} \times 19 \times 7 = \frac{133}{2} \text{ cm}^2 \]

4. **Calculate the Total Area of Quadrilateral ABCD:**
- Now, we add the areas of triangles ABC and ADC:
\[ \text{Area}_{ABCD} = \text{Area}_{ABC} + \text{Area}_{ADC} \]
\[ \text{Area}_{ABCD} = \frac{95}{2} + \frac{133}{2} = \frac{228}{2} = 114 \text{ cm}^2 \]

### Final Answer:
The area of quadrilateral ABCD is \(114 \text{ cm}^2\).

---