To solve the problem, we need to find the distance between the centroids of triangles PQR and DEF, where triangle DEF is formed by joining the midpoints of the sides of triangle ABC, and triangle PQR is formed by joining the midpoints of the sides of triangle DEF.
### Step-by-Step Solution:
1. **Identify the Centroid of Triangle ABC**:
- The centroid (G) of a triangle is the point where the three medians intersect. It divides each median into a ratio of 2:1.
- For triangle ABC with sides 4 cm, 5 cm, and 6 cm, we can find the coordinates of the centroid using the formula:
\[
G_{ABC} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
- However, we can skip the exact coordinates since we will be working with midpoints.
2. **Find the Midpoints of Triangle ABC**:
- Let’s denote the vertices of triangle ABC as A(0, 0), B(4, 0), and C(2, 5) (based on the given sides).
- The midpoints D, E, and F of sides AB, BC, and AC respectively can be calculated as follows:
- D = Midpoint of AB = \(\left( \frac{0 + 4}{2}, \frac{0 + 0}{2} \right) = (2, 0)\)
- E = Midpoint of BC = \(\left( \frac{4 + 2}{2}, \frac{0 + 5}{2} \right) = (3, 2.5)\)
- F = Midpoint of AC = \(\left( \frac{0 + 2}{2}, \frac{0 + 5}{2} \right) = (1, 2.5)\)
3. **Identify the Centroid of Triangle DEF**:
- The centroid (G) of triangle DEF can be found using the midpoints D, E, and F:
\[
G_{DEF} = \left( \frac{2 + 3 + 1}{3}, \frac{0 + 2.5 + 2.5}{3} \right) = \left( \frac{6}{3}, \frac{5}{3} \right) = (2, \frac{5}{3})
\]
4. **Find the Midpoints of Triangle DEF to Form Triangle PQR**:
- The midpoints of sides DE, EF, and FD can be calculated:
- P = Midpoint of DE = \(\left( \frac{2 + 3}{2}, \frac{0 + 2.5}{2} \right) = \left( 2.5, 1.25 \right)\)
- Q = Midpoint of EF = \(\left( \frac{3 + 1}{2}, \frac{2.5 + 2.5}{2} \right) = \left( 2, 2.5 \right)\)
- R = Midpoint of FD = \(\left( \frac{1 + 2}{2}, \frac{2.5 + 0}{2} \right) = \left( 1.5, 1.25 \right)\)
5. **Identify the Centroid of Triangle PQR**:
- The centroid (G) of triangle PQR can be found using the midpoints P, Q, and R:
\[
G_{PQR} = \left( \frac{2.5 + 2 + 1.5}{3}, \frac{1.25 + 2.5 + 1.25}{3} \right) = \left( \frac{6}{3}, \frac{5}{3} \right) = (2, \frac{5}{3})
\]
6. **Calculate the Distance Between Centroids G_{DEF} and G_{PQR}**:
- Since both centroids G_{DEF} and G_{PQR} are at the same coordinates (2, \(\frac{5}{3}\)), the distance between them is:
\[
\text{Distance} = \sqrt{(2 - 2)^2 + \left(\frac{5}{3} - \frac{5}{3}\right)^2} = \sqrt{0 + 0} = 0
\]
### Final Answer:
The distance between the centroids of triangles PQR and DEF is **0 units**.
