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To find the value of \( k \) in the given triangle, we can use similar triangles. Here's the step-by-step explanation:

### Step 1: Identify the triangles
The problem shows two right triangles:
1. The smaller triangle with height 5 and base \( k \).
2. The larger triangle with height 15 and base \( k + 60 \).

### Step 2: Set up the proportion
Since the triangles are similar (same angles), the ratios of their corresponding sides will be equal. Therefore, we can set up the proportion as follows:

\[
\frac{\text{Height of smaller triangle}}{\text{Height of larger triangle}} = \frac{\text{Base of smaller triangle}}{\text{Base of larger triangle}}
\]

Substituting the known values:

\[
\frac{5}{15} = \frac{k}{k + 60}
\]

### Step 3: Solve the proportion
Simplify the left-hand side:

\[
\frac{1}{3} = \frac{k}{k + 60}
\]

Now, cross-multiply:

\[
1 \cdot (k + 60) = 3 \cdot k
\]

This gives us:

\[
k + 60 = 3k
\]

Subtract \( k \) from both sides:

\[
60 = 2k
\]

Divide by 2:

\[
k = 30
\]

### Final Answer:
The value of \( k \) is \( 30 \), so the correct option is \( E \).

Would you like further details or explanations on this? Here are some related questions to consider:

1. How do you determine if two triangles are similar?
2. What other methods can be used to solve for unknown lengths in right triangles?
3. What are some real-world applications of similar triangles?
4. How would the proportion change if the height of the smaller triangle was different?
5. Can you solve the problem using trigonometric functions instead of similar triangles?

**Tip:** Always check for similar triangles in geometry problems, as they can simplify complex problems into manageable proportions.

Learn how to solve for the value of k in a geometry problem involving similar triangles. This step-by-step solution covers the use of proportions and the Similar Triangle Theorem. Suitable for students in Grades 8-10.

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