Here's the solution to your geometry problem: 1. Prove that $DE || FC$. Step 1: Calculate the ratio $\frac{AD}{DF}$. We are given that $\frac{AD}{AF} = \frac{3}{5}$. This implies that $AD$ represents $3$ parts of $AF$, and $AF$ represents $5$ parts. Therefore, $DF = AF - AD = 5 \text{ parts} - 3 \text{ parts} = 2 \text{ parts}$. So, the ratio $\frac{AD}{DF} = \frac{3}{2}$. Step 2: Calculate the ratio $\frac{AE}{EC}$. We are given $AE = 12$ units and $EC = 8$ units. $$ \frac{AE}{EC} = \frac{12}{8} = \frac{3}{2} $$ Step 3: Compare the ratios and conclude. Since $\frac{AD}{DF} = \frac{AE}{EC} = \frac{3}{2}$, by the converse of the Basic Proportionality Theorem (also known as Thales's Theorem), the line segment $DE$ is parallel to the line segment $FC$. Therefore, $\boxed{DE || FC}$. 2. If $AB = 14$ units, calculate the length of $BF$. Step 1: Interpret the parallel line statement. The statement "E on AC // FE" is interpreted to mean that the line segment $FE$ is parallel to the side $BC$ of $\triangle ABC$. So, $FE || BC$. Step 2: Apply Thales's Theorem (Basic Proportionality Theorem) to $\triangle ABC$. Since $FE || BC$, we can write the proportion: $$ \frac{AF}{AB} = \frac{AE}{AC} $$ Step 3: Calculate the length of $AC$. $AC$ is the sum of $AE$ and $EC$. $$ AC = AE + EC = 12 \text{ units} + 8 \text{ units} = 20 \text{ units} $$ Step 4: Substitute the known values into the proportion to find $AF$. We have $AB = 14$ units, $AE = 12$ units, and $AC = 20$ units. $$ \frac{AF}{14} = \frac{12}{20} $$ To solve for $AF$, multiply both sides by $14$: $$ AF = 14 \times \frac{12}{20} $$ Simplify the fraction $\frac{12}{20}$ to $\frac{3}{5}$: $$ AF = 14 \times \frac{3}{5} $$ $$ AF = \frac{42}{5} $$ $$ AF = 8.4 \text{ units} $$ Step 5: Calculate the length of $BF$. The length of $BF$ is the total length of $AB$ minus the length of $AF$. $$ BF = AB - AF $$ $$ BF = 14 \text{ units} - 8.4 \text{ units} $$ $$ BF = 5.6 \text{ units} $$ The length of $BF$ is $\boxed{5.6 \text{ units}}$. Send me the next one 📸