D and E are two points on the sides AC and BC respectively of `DeltaABC` such that `DE= 18 cm, CE = 5 cm` and `/_DEC = 90^@`. If tan `/_ABC = 3.6`, then `AC : CD = `

Options

A. `BC: 2 CE`
B. `2 CE: BC`
C. `2 BC: CE`
D. `CE: 2 BC`

Answer

To solve the problem, we need to find the ratio \( AC : CD \) given the following information:

- \( DE = 18 \, \text{cm} \)
- \( CE = 5 \, \text{cm} \)
- \( \angle DEC = 90^\circ \)
- \( \tan \angle ABC = 3.6 \)

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have triangle \( ABC \) with points \( D \) and \( E \) on sides \( AC \) and \( BC \) respectively. We know that \( DE \) is perpendicular to \( CE \).

2. **Setting Up the Right Triangle**:
Since \( \angle DEC = 90^\circ \), we can use the right triangle \( DEC \) to find the lengths of the sides. We know:
- \( DE = 18 \, \text{cm} \)
- \( CE = 5 \, \text{cm} \)

3. **Finding \( DC \)**:
Using the Pythagorean theorem in triangle \( DEC \):
\[ DC = \sqrt{DE^2 + CE^2} = \sqrt{18^2 + 5^2} = \sqrt{324 + 25} = \sqrt{349} \approx 18.7 \, \text{cm} \]

4. **Using the Tangent**:
We know that \( \tan \angle ABC = 3.6 \). The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, we can express it as:
\[ \tan \angle ABC = \frac{DE}{AC} = 3.6 \]
Rearranging gives us:
\[ AC = \frac{DE}{\tan \angle ABC} = \frac{18}{3.6} = 5 \, \text{cm} \]

5. **Finding \( AC : CD \)**:
Now we have:
- \( AC = 5 \, \text{cm} \)
- \( CD = DC - CE = 18.7 - 5 = 13.7 \, \text{cm} \)

The ratio \( AC : CD \) can be calculated as:
\[ AC : CD = 5 : 13.7 \]

6. **Simplifying the Ratio**:
To express this ratio in simpler terms, we can multiply both sides by 10 to avoid decimals:
\[ 5 \times 10 : 13.7 \times 10 = 50 : 137 \]

### Final Answer:
The ratio \( AC : CD = 50 : 137 \).