If D and E are points on the sides AB and AC respectively of a triangle ABC such that DE || BC. If AD =x cm, DB =(x-3) cm, AE = ( x+ 3 ) cm and EC = ( x -2) cm, what is the value (in cm. ) of x ?

Options

A. 3
B. 3.5
C. 4
D. 4.5

Answer

To solve the problem step by step, we will use the properties of similar triangles and the Basic Proportionality Theorem (BPT), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.

### Step 1: Set up the proportions using BPT
Since DE is parallel to BC, we can set up the following proportion based on the segments created on sides AB and AC:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]

### Step 2: Substitute the given lengths into the proportion
We know:
- \(AD = x\)
- \(DB = x - 3\)
- \(AE = x + 3\)
- \(EC = x - 2\)

Substituting these values into the proportion gives us:
\[ \frac{x}{x - 3} = \frac{x + 3}{x - 2} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ x \cdot (x - 2) = (x - 3) \cdot (x + 3) \]

### Step 4: Expand both sides
Expanding both sides:
- Left side: \(x^2 - 2x\)
- Right side: Using the difference of squares formula, we have \((x - 3)(x + 3) = x^2 - 9\)

So, we can write:
\[ x^2 - 2x = x^2 - 9 \]

### Step 5: Simplify the equation
Subtract \(x^2\) from both sides:
\[ -2x = -9 \]

### Step 6: Solve for \(x\)
Dividing both sides by -1 gives:
\[ 2x = 9 \]
Now, divide by 2:
\[ x = \frac{9}{2} = 4.5 \]

### Conclusion
The value of \(x\) is \(4.5\) cm.

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