An image of similar triangles is shown. Use the image to answer the questions.

a triangle GEH with side GE labeled 15, side GH labeled 12, and side HE labeled 9 and a second triangle DEF with side DF labeled 16

Part A: Find the length of side FE. Show each step of your work. (2 points)

Part B: Find the length of side DE. Show each step of your work. (2 points)

To solve for the missing side lengths in the given similar triangles, we will use the fact that corresponding sides of similar triangles are proportional.

### Step 1: Understanding the given data

- Triangle \( GEH \):
  - Side \( GE = 15 \)
  - Side \( GH = 12 \)
  - Side \( HE = 9 \)

- Triangle \( DEF \) (similar to triangle \( GEH \)):
  - Side \( DF = 16 \)
  - We need to find the length of sides \( FE \) and \( DE \).

### Part A: Find the length of side \( FE \)

Since the triangles are similar, we can set up a proportion between corresponding sides of triangle \( GEH \) and triangle \( DEF \).

Corresponding sides:
- \( GE \) corresponds to \( DE \)
- \( HE \) corresponds to \( FE \)
- \( GH \) corresponds to \( DF \)

Now, we use the proportion:

\[
\frac{HE}{FE} = \frac{GH}{DF}
\]

Substitute the known values:

\[
\frac{9}{FE} = \frac{12}{16}
\]

Solve for \( FE \):

\[
\frac{9}{FE} = \frac{3}{4}
\]

Now cross multiply:

\[
9 \times 4 = 3 \times FE
\]

\[
36 = 3 \times FE
\]

Now divide by 3:

\[
FE = \frac{36}{3} = 12
\]

Thus, the length of side \( FE \) is **12** units.

### Part B: Find the length of side \( DE \)

Next, we use the proportion for the corresponding sides \( GE \) and \( DE \):

\[
\frac{GE}{DE} = \frac{GH}{DF}
\]

Substitute the known values:

\[
\frac{15}{DE} = \frac{12}{16}
\]

Simplify the ratio on the right-hand side:

\[
\frac{15}{DE} = \frac{3}{4}
\]

Now cross multiply:

\[
15 \times 4 = 3 \times DE
\]

\[
60 = 3 \times DE
\]

Divide by 3:

\[
DE = \frac{60}{3} = 20
\]

Thus, the length of side \( DE \) is **20** units.

### Final Answers:
- **Part A:** Length of side \( FE = 12 \) units.
- **Part B:** Length of side \( DE = 20 \) units.

Would you like more details or have any further questions?

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Here are 5 questions to extend this concept:
1. How do you find the area of each triangle if given only the sides?
2. What if the sides were labeled differently; how would that change your proportions?
3. How can you prove that two triangles are similar beyond comparing their sides?
4. What happens to the ratios if you scale one triangle larger or smaller?
5. How can the Pythagorean Theorem be applied in these types of triangle problems?

**Tip:** Always double-check which sides are corresponding in similar triangles to ensure your proportions are set up correctly.

Learn how to solve a problem involving similar triangles GEH and DEF by finding the lengths of missing sides FE and DE using proportions. Step-by-step solution provided. Suitable for students in Grades 8-10.

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