In the given figure, if DE || BC, AE = 8 cm, EC = 2 cm and BC = 6 cm, then find DE

Answer

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

### Step 1: Identify the triangles
We have two triangles, triangle ADE and triangle ABC. Since DE is parallel to BC, we can use the properties of similar triangles.

### Step 2: Establish similarity
Since DE || BC, we know that:
- Angle DAE = Angle BAC (common angle)
- Angle ADE = Angle ABC (corresponding angles)

Thus, by the Angle-Angle (AA) criterion for similarity, triangle ADE is similar to triangle ABC.

### Step 3: Set up the proportion
For similar triangles, the ratios of corresponding sides are equal. Therefore, we can write the following proportion:
\[ \frac{AE}{EC} = \frac{DE}{BC} \]

### Step 4: Substitute known values
We know:
- AE = 8 cm
- EC = 2 cm
- BC = 6 cm

Substituting these values into the proportion gives us:
\[ \frac{8}{2} = \frac{DE}{6} \]

### Step 5: Simplify the left side
Calculating the left side:
\[ \frac{8}{2} = 4 \]
So, we have:
\[ 4 = \frac{DE}{6} \]

### Step 6: Cross-multiply to solve for DE
Cross-multiplying gives us:
\[ 4 \times 6 = DE \]
\[ DE = 24 \]

### Step 7: Solve for DE
Now, we need to find DE in terms of the total length AE + EC:
\[ AE + EC = 8 + 2 = 10 \]
Now we can set up the proportion again:
\[ \frac{DE}{BC} = \frac{DE}{6} = \frac{8}{10} \]

### Step 8: Solve for DE
Cross-multiplying gives us:
\[ 10 \cdot DE = 48 \]
\[ DE = \frac{48}{10} = 4.8 \text{ cm} \]

### Final Answer
Thus, the length of DE is:
\[ DE = 4.8 \text{ cm} \]

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