Area of a quadrilateral ABCD is 20 `cm^2` and perpendiculars on BD from opposite vertices are 1 cm and 1.5 cm. The length of BD is

Options

A. 4 cm
B. 15 cm
C. 16 cm
D. 18 cm

Answer

To find the length of diagonal BD in quadrilateral ABCD, we can use the formula for the area of a quadrilateral when the perpendiculars from opposite vertices to the diagonal are known.

### Step-by-Step Solution:

1. **Identify the given values:**
- Area of quadrilateral ABCD = 20 cm²
- Perpendicular from vertex A to BD (height from A) = 1 cm
- Perpendicular from vertex C to BD (height from C) = 1.5 cm

2. **Use the formula for the area of a quadrilateral:**
The area of quadrilateral ABCD can be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{BD} \times (\text{Height from A} + \text{Height from C}) \]
Substituting the known values:
\[ 20 = \frac{1}{2} \times \text{BD} \times (1 + 1.5) \]

3. **Calculate the sum of the heights:**
\[ 1 + 1.5 = 2.5 \]
Now, substituting this back into the area formula:
\[ 20 = \frac{1}{2} \times \text{BD} \times 2.5 \]

4. **Simplify the equation:**
Multiply both sides by 2 to eliminate the fraction:
\[ 40 = \text{BD} \times 2.5 \]

5. **Solve for BD:**
Divide both sides by 2.5:
\[ \text{BD} = \frac{40}{2.5} \]

6. **Perform the division:**
\[ \text{BD} = 16 \text{ cm} \]

### Final Answer:
The length of diagonal BD is **16 cm**.