For a triangle ABC,D and E are two points on AB and AC such that `AD = 1/4 AB, AE = 1/4 AC`. If `BC = 12 cm`, then `DE` is

Options

A. a) 5 cm
B. b) 4 cm
C. c) 3 cm
D. d) 6 cm

Answer

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

### Step 1: Understand the Triangle and Points
We have triangle ABC with points D and E on sides AB and AC respectively such that:
- \( AD = \frac{1}{4} AB \)
- \( AE = \frac{1}{4} AC \)

### Step 2: Set Up Ratios Using Similar Triangles
Since D and E divide sides AB and AC in the same ratio, triangles ADE and ABC are similar. Therefore, the ratios of corresponding sides are equal:
\[ \frac{AD}{AB} = \frac{DE}{BC} = \frac{AE}{AC} \]

### Step 3: Substitute Known Values
From the problem, we know:
- \( BC = 12 \, \text{cm} \)
- \( AD = \frac{1}{4} AB \)
- \( AE = \frac{1}{4} AC \)

### Step 4: Express Ratios
Using the similarity of triangles, we can express the ratios:
\[ \frac{AD}{AB} = \frac{1}{4} \]
This implies:
\[ \frac{DE}{BC} = \frac{1}{4} \]

### Step 5: Solve for DE
Now we can substitute the known value of BC into the ratio:
\[ \frac{DE}{12} = \frac{1}{4} \]

To find DE, we cross-multiply:
\[ DE = \frac{1}{4} \times 12 \]
\[ DE = 3 \, \text{cm} \]

### Conclusion
Thus, the length of DE is \( 3 \, \text{cm} \).

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