Here's how to find the length of DE. Step 1: Identify similar triangles. Given that BA is parallel to CE, we can identify two similar triangles: $\triangle DCE$ and $\triangle DBA$. • $\angle CDE = \angle BDA$ (Common angle) • $\angle CED = \angle BAD$ (Corresponding angles, since CE || BA and AD is a transversal) • $\angle DCE = \angle DBA$ (Corresponding angles, since CE || BA and BD is a transversal) Therefore, $\triangle DCE \sim \triangle DBA$ by the Angle-Angle-Angle (AAA) similarity criterion. Step 2: Set up the ratio of corresponding sides. From the similarity $\triangle DCE \sim \triangle DBA$, the ratio of corresponding sides is equal: $$ \frac{CE}{BA} = \frac{DE}{DA} $$ Step 3: Substitute the given values. We are given: BA = $9 \text{ cm}$ CE = $4 \text{ cm}$ AE = $3 \text{ cm}$ Let DE = $x$. Then DA = DE + AE = $x + 3 \text{ cm}$. Substitute these values into the ratio: $$ \frac{4}{9} = \frac{x}{x+3} $$ Step 4: Solve for $x$. Cross-multiply the equation: $$ 4(x+3) = 9x $$ Distribute the $4$ on the left side: $$ 4x + 12 = 9x $$ Subtract $4x$ from both sides: $$ 12 = 9x - 4x $$ $$ 12 = 5x $$ Divide by $5$: $$ x = \frac{12}{5} $$ $$ x = 2.4 $$ So, the length of DE is $2.4 \text{ cm}$. The length of DE is $\boxed{\text{2.4 cm}}$. 3 done, 2 left today. You're making progress.