We are given that the area of triangle $\triangle ABC$ is 64, and $P, Q, R$ are the midpoints of sides $AB$, $AC$, and $BC$ respectively. $X, Y, Z$ are the midpoints of the sides of triangle $PQR$. The task is to find the area of triangle $XYZ$.

Step-by-step Solution:

1. Understand the relation between the areas of triangles:

   • When midpoints of a triangle are connected, the triangle formed is similar to the original triangle, and its area is reduced by a factor of $4$. Specifically, the area of the triangle formed by connecting the midpoints of a triangle is $\frac{1}{4}$ of the original triangle.

2. Apply the area reduction for triangle $PQR$:

   • Triangle $PQR$ is formed by joining the midpoints of triangle $ABC$. Therefore, the area of triangle $PQR$ is $\frac{1}{4}$ of the area of triangle $ABC$.
   • Area of triangle $PQR$ = $\frac{1}{4} \times 64 = 16$.

3. Apply the area reduction for triangle $XYZ$:

   • Triangle $XYZ$ is formed by joining the midpoints of triangle $PQR$. Similarly, the area of triangle $XYZ$ is $\frac{1}{4}$ of the area of triangle $PQR$.
   • Area of triangle $XYZ$ = $\frac{1}{4} \times 16 = 4$.

Thus, the area of triangle $XYZ$ is $\boxed{4}$.

Would you like further clarification or details on any part? Here are 5 related questions to explore:

1. How do midpoints affect the similarity and area ratio of triangles?
2. Can the area of any triangle be found using midpoints alone?
3. What is the significance of similar triangles in geometric problems?
4. How is the area ratio of similar triangles related to the side ratio?
5. How do you prove that the triangle formed by midpoints has $\frac{1}{4}$ of the area?

Tip: Always remember that when you connect midpoints of a triangle, you form a triangle that is similar to the original one, with its area reduced by a factor of 4!