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!