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Two similar triangles are given.  Find the proportionality constant k such that the length of each side of the larger triangle is k times the length of the corresponding sides of the smaller triangle.  Then find the lengths of the missing sides of each triangle.

To solve this problem, we need to determine the proportionality constant \( k \) and then use it to find the missing sides in both triangles.

### Step 1: Identify corresponding sides
The triangles \( \triangle GKH \) and \( \triangle TCQ \) are similar because their corresponding angles are equal (as indicated by the markings). The corresponding sides are proportional, so we can write the following ratios:

- \( \frac{GH}{TQ} = \frac{KH}{CQ} = \frac{GK}{TC} = k \)

### Step 2: Calculate the proportionality constant \( k \)
Using the known sides:

\[
k = \frac{GH}{TQ} = \frac{17}{10}
\]

\[
k = 1.7
\]

### Step 3: Find the missing sides
Now that we have \( k = 1.7 \), we can find the missing sides.

#### For triangle \( GKH \):
- Given \( GK = 6 \), and we need to find \( KH \):
  
\[
KH = CQ \times k = 5 \times 1.7 = 8.5
\]

So, \( KH = 8.5 \).

#### For triangle \( TCQ \):
- \( TQ = 10 \) (already known), and \( CQ = 5 \),
- \( TC \) corresponds to \( GK \):

\[
TC = \frac{GK}{k} = \frac{6}{1.7} \approx 3.53
\]

So, \( TC \approx 3.53 \).

### Summary of Results:
- Proportionality constant \( k = 1.7 \).
- Missing side \( KH = 8.5 \) in triangle \( GKH \).
- Missing side \( TC \approx 3.53 \) in triangle \( TCQ \).

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How can you use the proportionality constant in real-life applications?
2. What other methods could you use to verify the similarity of triangles?
3. Can the proportionality constant \( k \) ever be less than 1? What would that imply?
4. How does the triangle inequality theorem apply to the sides of these triangles?
5. What are the implications of triangle similarity in coordinate geometry?

**Tip:** When dealing with similar triangles, always verify that corresponding angles are equal to ensure the triangles are indeed similar. This will help in setting up correct proportions.

Learn how to find the proportionality constant and missing sides of similar triangles using a step-by-step solution. This problem covers key concepts in geometry, including triangle similarity and proportionality. Suitable for high school students.

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