In the figure, EF || AC, BC = 10 cm, AB = 13 cm and EC = 2 cm, find AF

Answer

To solve the problem step by step, we will use the information given and apply Thales' theorem.

### Step 1: Understand the Given Information
We have a figure where EF is parallel to AC. The lengths given are:
- BC = 10 cm
- AB = 13 cm
- EC = 2 cm

We need to find the length of AF.

### Step 2: Calculate BE
Since EF is parallel to AC, we can find the length of BE using the relation:
\[ BE = BC - EC \]
Substituting the values:
\[ BE = 10 \, \text{cm} - 2 \, \text{cm} = 8 \, \text{cm} \]

### Step 3: Set Up the Variables
Let:
- AF = x cm
- BF = AB - AF = 13 cm - x cm

### Step 4: Apply Thales' Theorem
According to Thales' theorem, the ratios of the segments are equal:
\[ \frac{BF}{AF} = \frac{BE}{EC} \]
Substituting the known values:
\[ \frac{13 - x}{x} = \frac{8}{2} \]
This simplifies to:
\[ \frac{13 - x}{x} = 4 \]

### Step 5: Cross-Multiply to Solve for x
Cross-multiplying gives:
\[ 13 - x = 4x \]
Now, rearranging the equation:
\[ 13 = 4x + x \]
\[ 13 = 5x \]

### Step 6: Solve for x
Dividing both sides by 5:
\[ x = \frac{13}{5} \]
Calculating this gives:
\[ x = 2.6 \, \text{cm} \]

### Step 7: Conclusion
Thus, the length of AF is:
\[ AF = 2.6 \, \text{cm} \]