A △DEF is formed by joining the mid-points of the sides of △ ABC. Similarly, a △ PQR is formed by joining the mid-points of the sides of the △ DEF. If the sides of the △ PQR are of lengths `1, 2` and `3` units, what is the perimeter of the AABC?

Options

A. 18 units
B. 24 units
C. 48 units
D. Cannot be determined

Answer

To find the perimeter of triangle ABC given that triangle PQR is formed by joining the midpoints of triangle DEF, we can follow these steps:

### Step 1: Understand the relationship between the triangles
When we form triangle DEF by joining the midpoints of triangle ABC, each side of triangle DEF is half the length of the corresponding side of triangle ABC. Similarly, when we form triangle PQR by joining the midpoints of triangle DEF, each side of triangle PQR is half the length of the corresponding side of triangle DEF.

### Step 2: Define the sides of triangle PQR
Given that the sides of triangle PQR are 1, 2, and 3 units, we can denote these sides as:
- Side PQ = 1 unit
- Side QR = 2 units
- Side RP = 3 units

### Step 3: Calculate the perimeter of triangle PQR
The perimeter of triangle PQR (P3) can be calculated as:
\[ P3 = PQ + QR + RP = 1 + 2 + 3 = 6 \text{ units} \]

### Step 4: Relate the perimeter of triangle PQR to triangle DEF
Since triangle PQR is formed by the midpoints of triangle DEF, the perimeter of triangle DEF (P2) will be double that of triangle PQR:
\[ P2 = 2 \times P3 = 2 \times 6 = 12 \text{ units} \]

### Step 5: Relate the perimeter of triangle DEF to triangle ABC
Similarly, since triangle DEF is formed by the midpoints of triangle ABC, the perimeter of triangle ABC (P1) will be double that of triangle DEF:
\[ P1 = 2 \times P2 = 2 \times 12 = 24 \text{ units} \]

### Conclusion
Thus, the perimeter of triangle ABC is:
\[ \text{Perimeter of ABC} = 24 \text{ units} \]