In `DeltaABC, DEabs()BC`. If AD= 2.4 cm, AE= 3.2 cm, DE= 2 cm and BC= 5 cm, find BD and CE.

Answer

To solve the problem, we will use the properties of similar triangles since DE is parallel to BC. Let's break down the solution step by step.

### Step 1: Understand the given information
We have triangle ABC with DE parallel to BC. The lengths provided are:
- AD = 2.4 cm
- AE = 3.2 cm
- DE = 2 cm
- BC = 5 cm

We need to find the lengths of BD and CE.

### Step 2: Set up the proportion using similar triangles
Since DE is parallel to BC, triangles ADE and ABC are similar. Therefore, we can set up the following proportion:

\[ \frac{AD}{AB} = \frac{DE}{BC} \]

Let BD = x and CE = y. Then, we can express AB and AC in terms of x and y:
- AB = AD + BD = 2.4 + x
- AC = AE + CE = 3.2 + y

### Step 3: Substitute the known values into the proportion
Substituting the known values into the proportion gives us:

\[ \frac{2.4}{2.4 + x} = \frac{2}{5} \]

### Step 4: Cross-multiply to solve for x
Cross-multiplying yields:

\[ 2.4 \cdot 5 = 2 \cdot (2.4 + x) \]

This simplifies to:

\[ 12 = 4.8 + 2x \]

### Step 5: Solve for x
Now, isolate x:

\[ 12 - 4.8 = 2x \]
\[ 7.2 = 2x \]
\[ x = \frac{7.2}{2} = 3.6 \text{ cm} \]

So, BD = 3.6 cm.

### Step 6: Find CE using another proportion
Now, we can use the other proportion to find CE:

\[ \frac{AE}{AC} = \frac{DE}{BC} \]

Substituting the known values gives us:

\[ \frac{3.2}{3.2 + y} = \frac{2}{5} \]

### Step 7: Cross-multiply to solve for y
Cross-multiplying yields:

\[ 3.2 \cdot 5 = 2 \cdot (3.2 + y) \]

This simplifies to:

\[ 16 = 6.4 + 2y \]

### Step 8: Solve for y
Now, isolate y:

\[ 16 - 6.4 = 2y \]
\[ 9.6 = 2y \]
\[ y = \frac{9.6}{2} = 4.8 \text{ cm} \]

So, CE = 4.8 cm.

### Final Answer
Thus, the lengths are:
- BD = 3.6 cm
- CE = 4.8 cm

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