In `Delta ABC and Delta DEF`, it is given that AB = 5 cm, BC = 4 cm, CA = 4.2 cm, DE = 10 cm, EF = 8 cm and FD = 8.4 cm. If AL is perpendicular to BC and DM is perpendicular to EF, then what is the ratio of AL to DM.

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. 1

Answer

To solve the problem, we need to find the ratio of the altitudes \( AL \) and \( DM \) in triangles \( ABC \) and \( DEF \) respectively. We will use the properties of similar triangles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- For triangle \( ABC \):
- \( AB = 5 \, \text{cm} \)
- \( BC = 4 \, \text{cm} \)
- \( CA = 4.2 \, \text{cm} \)
- For triangle \( DEF \):
- \( DE = 10 \, \text{cm} \)
- \( EF = 8 \, \text{cm} \)
- \( FD = 8.4 \, \text{cm} \)

2. **Draw the Triangles:**
- Draw triangle \( ABC \) and triangle \( DEF \).
- Mark the points \( A, B, C \) for triangle \( ABC \) and \( D, E, F \) for triangle \( DEF \).
- Draw altitude \( AL \) from point \( A \) to line \( BC \) and altitude \( DM \) from point \( D \) to line \( EF \).

3. **Set Up the Ratios of Corresponding Sides:**
- We will compare the sides of the triangles:
\[ \frac{AB}{DE}, \quad \frac{BC}{EF}, \quad \frac{CA}{FD} \]

4. **Calculate the Ratios:**
- Calculate each ratio:
- \( \frac{AB}{DE} = \frac{5}{10} = \frac{1}{2} \)
- \( \frac{BC}{EF} = \frac{4}{8} = \frac{1}{2} \)
- \( \frac{CA}{FD} = \frac{4.2}{8.4} = \frac{1}{2} \)

5. **Conclude Similarity of Triangles:**
- Since all three ratios are equal:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{1}{2} \]
- By the Side-Side-Side (SSS) similarity criterion, triangles \( ABC \) and \( DEF \) are similar.

6. **Use the Property of Similar Triangles:**
- The ratio of the altitudes of similar triangles is equal to the ratio of their corresponding sides.
- Therefore, the ratio of the altitudes \( AL \) and \( DM \) is:
\[ \frac{AL}{DM} = \frac{1}{2} \]

### Final Answer:
The ratio of \( AL \) to \( DM \) is \( \frac{1}{2} \).

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