The given problem is about the dilation of triangle $XYZ$ with center $P$ to form triangle $X'Y'Z'$. Dilation is a transformation that changes the size of a figure but preserves the shape and the proportional relationships of the sides.

Let's analyze the options:

1. Option 1: $X'X = Y'Y = Z'Z$

   • This suggests that the distances from the vertices of the original triangle to their corresponding vertices in the enlarged triangle are equal. However, this is incorrect, as the distances from the center of dilation $P$ to each point depend on the scaling factor and are not necessarily equal.

2. Option 2: $\frac{XY}{X'Y'} = \frac{YZ}{Y'Z'} = \frac{XZ}{X'Z'}$

   • This option implies that the ratios of corresponding sides of the triangles are equal, which is a correct characteristic of dilation. The sides of the original and enlarged triangle are proportional based on the scale factor of the dilation. This statement is true.

3. Option 3: $\triangle X'Y'Z' \cong \triangle XYZ$

   • This option suggests that the two triangles are congruent. However, dilation only preserves the shape and not the size, so the triangles are similar, not congruent. Thus, this is incorrect.

4. Option 4: $\frac{YZ}{Y'Z'} = \frac{XZ}{X'Y'} = \frac{XY}{X'Z'}$

   • This option incorrectly mixes up the corresponding sides, so this is incorrect.

Correct Answer:

The second option is the correct conclusion: $\frac{XY}{X'Y'} = \frac{YZ}{Y'Z'} = \frac{XZ}{X'Z'}$

Would you like further clarification or details on this?

Here are 5 related questions:

1. What is the difference between congruence and similarity in triangles?
2. How is the scale factor of a dilation calculated?
3. What properties are preserved during a dilation?
4. How does the center of dilation affect the positions of the enlarged triangle’s vertices?
5. Can dilation result in a smaller triangle? If so, how?

Tip: In dilation, the corresponding angles of the original and dilated figures remain equal, but the sides are proportional based on the scale factor.