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The problem presented shows a right triangle \( \triangle DEF \), with side \( f = 5 \, \text{cm} \), hypotenuse \( e = 13 \, \text{cm} \), and asks us to find the length of side \( d = EF \).

To solve for \( d \), we can use the **Pythagorean theorem**:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

In this case:
- \( f = 5 \, \text{cm} \) (one of the legs),
- \( d = EF \) (the other leg, which we need to find),
- \( e = 13 \, \text{cm} \) (the hypotenuse).

### Step-by-Step Solution

1. Write down the Pythagorean theorem for \( \triangle DEF \):
   \[
   d^2 + f^2 = e^2
   \]

2. Substitute the given values:
   \[
   d^2 + 5^2 = 13^2
   \]

3. Calculate \( 5^2 \) and \( 13^2 \):
   \[
   d^2 + 25 = 169
   \]

4. Isolate \( d^2 \) by subtracting 25 from both sides:
   \[
   d^2 = 169 - 25
   \]
   \[
   d^2 = 144
   \]

5. Take the square root of both sides to solve for \( d \):
   \[
   d = \sqrt{144}
   \]
   \[
   d = 12 \, \text{cm}
   \]

### Answer
The length of \( EF \) is \( \mathbf{12 \, \text{cm}} \).

Would you like further details on this solution, or have any questions?

Here are five related questions that might expand your understanding:
1. How would we solve this if only the angle opposite \( f \) was given, instead of both sides?
2. What other properties can we deduce about triangle \( DEF \) based on its side lengths?
3. How can we verify if a triangle is a right triangle by using side lengths alone?
4. What if \( e \) and \( d \) were given, and we needed to find \( f \)?
5. How does the Pythagorean theorem relate to trigonometric identities?

**Tip:** Always check the lengths provided in right triangles to see if they follow Pythagorean triples (like 5, 12, 13) for quicker verification.

Learn how to find the length of EF in a right triangle with given side lengths. This problem involves using the Pythagorean theorem to solve for the missing side, suitable for Grade 8 students.

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