A line segment DE is drawn parallel to base BC of A ABC which cuts AB at point D and AC at point E. If AB = 5 BD and EC = 3.2 cm, find the length of AE. 

Answer

To solve the problem step by step, we will use the properties of similar triangles.

### Step 1: Understand the relationship between segments
Given that \( AB = 5 \times BD \), we can express \( AD \) in terms of \( BD \):
\[ AD = AB - BD = 5BD - BD = 4BD \]

### Step 2: Set up the ratio of segments
Now we can set up the ratio of the segments \( AD \) and \( AB \):
\[ \frac{AD}{AB} = \frac{4BD}{5BD} = \frac{4}{5} \]

### Step 3: Use the property of similar triangles
Since \( DE \) is parallel to \( BC \), triangles \( ADE \) and \( ABC \) are similar by the AA (Angle-Angle) similarity criterion. Therefore, the ratios of corresponding sides are equal:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]

### Step 4: Express \( AC \) in terms of \( AE \)
We know that \( AC = AE + EC \). Given \( EC = 3.2 \) cm, we can substitute this into our ratio:
\[ \frac{AD}{AB} = \frac{AE}{AE + 3.2} \]

### Step 5: Substitute the known values into the ratio
Now, substituting the known values into the equation:
\[ \frac{4}{5} = \frac{AE}{AE + 3.2} \]

### Step 6: Cross-multiply to solve for \( AE \)
Cross-multiplying gives:
\[ 4(AE + 3.2) = 5AE \]
Expanding this:
\[ 4AE + 12.8 = 5AE \]

### Step 7: Rearranging the equation
Rearranging the equation to isolate \( AE \):
\[ 12.8 = 5AE - 4AE \]
\[ 12.8 = AE \]

### Step 8: Conclusion
Thus, the length of \( AE \) is:
\[ AE = 12.8 \, \text{cm} \]

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