`Delta ABC and Delta DEF `are similar. Also `/_A = /_D and /_B = /_E`. If `4AB =DE `and `BC = 12 cm`, then EF is equal to

Options

A. 3 cm
B. 24 cm
C. 16 cm
D. 48 cm

Answer

To solve the problem, we need to use the properties of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the similarity of triangles
Given that triangles ABC and DEF are similar, we know that the ratios of their corresponding sides are equal. This means:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \]

### Step 2: Use the given information
We are given that:
1. \( 4AB = DE \)
2. \( BC = 12 \, \text{cm} \)

### Step 3: Express DE in terms of AB
From the equation \( 4AB = DE \), we can express \( DE \) as:
\[ DE = 4AB \]

### Step 4: Set up the ratio of corresponding sides
Using the similarity of triangles, we can set up the ratio:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Substituting \( DE \) from step 3 into the ratio gives us:
\[ \frac{AB}{4AB} = \frac{12}{EF} \]

### Step 5: Simplify the ratio
The left side simplifies to:
\[ \frac{1}{4} = \frac{12}{EF} \]

### Step 6: Cross-multiply to solve for EF
Cross-multiplying gives us:
\[ 1 \cdot EF = 4 \cdot 12 \]
\[ EF = 48 \, \text{cm} \]

### Conclusion
Thus, the length of \( EF \) is \( 48 \, \text{cm} \).

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