We have a triangle \( XYZ \) where:
- \( XY = 4x \),
- \( YZ = 3x \),
- \( XZ = 42 \).

Since this problem is likely dealing with a triangle, we can apply the triangle inequality or use the **Pythagorean theorem** if the triangle is right-angled. However, based on the information provided, let's explore if it's a right triangle by considering \( XZ \) as the hypotenuse and see if Pythagoras applies:

\[
XY^2 + YZ^2 = XZ^2
\]

Substitute the given values:

\[
(4x)^2 + (3x)^2 = 42^2
\]

\[
16x^2 + 9x^2 = 1764
\]

\[
25x^2 = 1764
\]

Now, solve for \( x \):

\[
x^2 = \frac{1764}{25} = 70.56
\]

\[
x = \sqrt{70.56} \approx 8.4
\]

Now, let's substitute \( x \approx 8.4 \) back into the expressions for the side lengths:

- \( XY = 4x = 4(8.4) \approx 33.6 \),
- \( YZ = 3x = 3(8.4) \approx 25.2 \).

Therefore, the side lengths of the triangle are approximately:
- \( XY \approx 33.6 \),
- \( YZ \approx 25.2 \),
- \( XZ = 42 \).

This confirms that the triangle is a right triangle.

### Do you want more details on any step, or do you have any questions?

Here are 5 related questions for further exploration:
1. How can you verify that a triangle is right-angled using only side lengths?
2. What is the triangle inequality, and how does it apply here?
3. If \( XZ \) was not the hypotenuse, would the solution change?
4. How would the problem change if \( XY = 5x \) and \( YZ = 2x \)?
5. Can you find the area of this triangle given these side lengths?

**Tip:** In any right triangle, you can always apply the Pythagorean theorem to relate the sides, but make sure that the triangle satisfies the conditions!

This problem explores a right triangle with side lengths XY = 4x, YZ = 3x, and hypotenuse XZ = 42. Learn how to use the Pythagorean theorem and algebraic substitution to find the value of x and the side lengths. Suitable for students in Grades 8-10.

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