https://file.aiquickdraw.com/mathbot/file/1727342265769u0ql45ht.jpg

The cuboid in the image has the following dimensions:

- \( DE = 12 \, \text{cm} \)
- \( BE = 13 \, \text{cm} \)
- \( EF = 3 \, \text{cm} \)

To calculate the length of \( FG \), we need to recognize that \( FG \) is a diagonal in the base of the cuboid formed by the vertices \( E, F, G, H \). 

### Step 1: Recognize the triangle
The points \( F, G, E \) form a right-angled triangle:
- \( EF = 3 \, \text{cm} \) (given),
- \( FG \) is what we need to calculate,
- \( EG \) is the diagonal of rectangle \( EFGH \), where \( EH = DE = 12 \, \text{cm} \).

### Step 2: Calculate diagonal \( EG \)
\( EG \) is the diagonal of a rectangle with sides \( EF \) and \( EH \). Using the Pythagorean theorem for right triangle \( EFG \):

\[
EG = \sqrt{EH^2 + EF^2}
\]

Substitute the values:

\[
EG = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \approx 12.37 \, \text{cm}
\]

### Step 3: Use Pythagorean theorem for triangle \( FEG \)
Now that we know \( EG \), we can find \( FG \). We apply the Pythagorean theorem again, considering the right triangle \( FEG \):

\[
FG^2 = EG^2 + BE^2
\]

Substitute the values:

\[
FG^2 = 12.37^2 + 13^2 = 153 + 169 = 322
\]

Finally, calculate \( FG \):

\[
FG = \sqrt{322} \approx 17.94 \, \text{cm}
\]

### Final Answer:
The length of \( FG \) is approximately **17.94 cm**.

---

Would you like more details or explanations on any steps? 

Here are 5 related questions to explore:
1. How would you calculate the diagonal of a cuboid?
2. Can the Pythagorean theorem be applied in three dimensions?
3. How would you calculate the surface area of this cuboid?
4. What is the formula for the space diagonal of a cuboid?
5. How can trigonometry be used to solve cuboid problems?

**Tip**: Always check if you're working with a right-angled triangle when applying the Pythagorean theorem, as it simplifies calculations.

ABCDEF is a cuboid. DE = 12cm, BE = 13cm, EF = 3cm, calculate the length of FG.

Learn how to calculate the diagonal length FG of a cuboid with given side lengths DE = 12cm, BE = 13cm, and EF = 3cm. The solution involves applying the Pythagorean theorem to multiple triangles. Suitable for students in Grades 9-10.

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