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The triangle in the image is a right triangle with one leg measuring 6 cm, the other leg measuring 12 cm, and an altitude from the right angle to the hypotenuse, with length \( x \). We are tasked with finding the value of \( x \).

### Step-by-Step Solution:

1. **Use of Geometric Mean Theorem (Altitude Theorem):**
   The altitude to the hypotenuse in a right triangle divides the triangle into two smaller right triangles. The length of the altitude \( x \) is the geometric mean of the two segments it divides the hypotenuse into.

   The hypotenuse of the original triangle is 6 cm + 12 cm = 18 cm.
   
   According to the Geometric Mean Theorem:
   \[
   x^2 = 6 \times 12
   \]
   \[
   x^2 = 72
   \]
   \[
   x = \sqrt{72}
   \]
   \[
   x = \sqrt{36 \times 2} = 6\sqrt{2}
   \]

### Conclusion:
The value of \( x \) is \( 6\sqrt{2} \), which corresponds to option **D**.

Would you like any further details or clarifications?

Here are 5 related questions to further your understanding:
1. How can the geometric mean theorem be applied to other types of triangles?
2. What is the importance of the Pythagorean Theorem in solving right triangles?
3. How does the altitude theorem relate to similar triangles?
4. How can we calculate the area of this triangle using the altitude?
5. What other theorems can be used to solve for unknown sides in right triangles?

**Tip:** For any right triangle, the geometric mean theorem provides a quick way to find the altitude if you know the lengths of the segments it divides on the hypotenuse.

What is the value of x in the given triangle with legs measuring 6 cm and 12 cm?

Solve for the altitude (x) in a right triangle where the legs measure 6 cm and 12 cm, using the Geometric Mean Theorem. Learn how to apply the theorem to calculate the altitude by finding the geometric mean of the two segments created by the altitude on the hypotenuse. Suitable for Grades 8-10 students.

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