Questions: In the figure to the right, triangle ABC and triangle ADE are similar. Find the length of EC.

Transcript text: In the figure to the right, $\triangle A B C$ and $\triangle A D E$ are similar. Find the length of $\overline{E C}$.

Solution

Solution Steps

Step 1: Identify Similar Triangles

Given that triangles $ \triangle ABC $ and $ \triangle ADE $ are similar, we can use the properties of similar triangles. Similar triangles have corresponding sides in proportion.

Step 2: Set Up Proportions

Since $ \triangle ABC $ and $ \triangle ADE $ are similar, the ratio of corresponding sides is equal. We can write the proportion as: \[ \frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE} \]

Step 3: Substitute Known Values

From the diagram:

$ AB = 1 $
$ AD = AB + BD = 1 + 9 = 10 $
$ BC = 1 $
$ DE = 14 $

Using the proportion: \[ \frac{AB}{AD} = \frac{BC}{DE} \] Substitute the known values: \[ \frac{1}{10} = \frac{1}{DE} \]

Step 4: Solve for EC

Since $ DE = 14 $, we need to find $ EC $. From the similar triangles, we know: \[ \frac{BC}{DE} = \frac{1}{14} \] Since $ DE = EC + 9 $, we can write: \[ \frac{1}{14} = \frac{1}{EC + 9} \]

Step 5: Solve the Equation

Cross-multiply to solve for $ EC $: \[ 1 \cdot (EC + 9) = 14 \cdot 1 \] \[ EC + 9 = 14 \] \[ EC = 14 - 9 \] \[ EC = 5 \]