A line DE parallel to the side BC intersects the other two sides of triangle at points D and E such that `AD = 1/6 AB` and `AE = 1/6 AC`. If the value of `BC `is 18 cm, calculate the value of `DE` (in cm.).

Options

A. a) 2
B. b) 3
C. c) 6
D. d) 8

Answer

To solve the problem, we will use the properties of similar triangles. Here are the steps:

### Step 1: Understand the triangle and the parallel line
We have triangle ABC with a line DE parallel to side BC. The line DE intersects sides AB and AC at points D and E, respectively.

### Step 2: Identify the given ratios
We know that:
- \( AD = \frac{1}{6} AB \)
- \( AE = \frac{1}{6} AC \)

### Step 3: Use the properties of similar triangles
Since DE is parallel to BC, triangles ADE and ABC are similar by the Angle-Angle (AA) similarity criterion. This gives us the following ratio:
\[ \frac{AD}{AB} = \frac{DE}{BC} = \frac{AE}{AC} \]

### Step 4: Substitute the known values
From the problem, we know:
- \( BC = 18 \, \text{cm} \)
- \( \frac{AD}{AB} = \frac{1}{6} \)

Using the similarity ratio:
\[ \frac{1}{6} = \frac{DE}{18} \]

### Step 5: Solve for DE
To find DE, we can cross-multiply:
\[ DE = \frac{1}{6} \times 18 \]
\[ DE = 3 \, \text{cm} \]

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

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