A riangle DEF is formed by joining the mid points of the sides of $ riangle ABC$. Similarly, a riangle PQR is formed by joining the midpoints of the sides of the $ riangle DEF$. If the sides of the $ riangle PQR$ are of lengths 4, 2 and 3 units, what is the perimeter of the $ riangle ABC$ ?

Q: A \triangle DEF is formed by joining the mid points of the sides of $\triangle ABC$. Similarly, a \triangle PQR is formed by joining the midpoints of the sides of the $\triangle DEF$. If the sides of the $\triangle PQR$ are of lengths 4, 2 and 3 units, what is the perimeter of the $\triangle ABC$ ?

A \triangle DEF is formed by joining the mid points of the sides of $\triangle ABC$. Similarly, a \triangle PQR is formed by joining the midpoints of the sides of the $\triangle DEF$. If the sides of the $\triangle PQR$ are of lengths 4, 2 and 3 units, what is the perimeter of the $\triangle ABC$ ?

Question

A 	riangle DEF is formed by joining the mid points of the sides of $	riangle ABC$. Similarly, a 	riangle PQR is formed by joining the midpoints of the sides of the $	riangle DEF$. If the sides of the $	riangle PQR$ are of lengths 4, 2 and 3 units, what is the perimeter of the $	riangle ABC$ ?

Accepted Answer

To determine the perimeter of $\triangle ABC$, we need to understand the relationship between the triangles formed by joining the midpoints of the sides of a triangle.

1. **Understanding the Relationship:**
   - When a triangle is formed by joining the midpoints of the sides of another triangle, the new triangle is known as the medial triangle. The sides of the medial triangle are parallel to the sides of the original triangle and are half the length of the corresponding sides of the original triangle.
   - Therefore, if $\triangle DEF$ is formed by joining the midpoints of $\triangle ABC$, then each side of $\triangle DEF$ is half the length of the corresponding side of $\triangle ABC$.
   - Similarly, if $\triangle PQR$ is formed by joining the midpoints of $\triangle DEF$, then each side of $\triangle PQR$ is half the length of the corresponding side of $\triangle DEF$.

2. **Calculating the Perimeter:**
   - Given that the sides of $\triangle PQR$ are 4, 2, and 3 units, the perimeter of $\triangle PQR$ is $4 + 2 + 3 = 9$ units.
   - Since $\triangle PQR$ is formed by joining the midpoints of $\triangle DEF$, each side of $\triangle DEF$ is twice the length of the corresponding side of $\triangle PQR$. Therefore, the sides of $\triangle DEF$ are $8, 4, $ and $6$ units, and its perimeter is $8 + 4 + 6 = 18$ units.
   - Similarly, since $\triangle DEF$ is formed by joining the midpoints of $\triangle ABC$, each side of $\triangle ABC$ is twice the length of the corresponding side of $\triangle DEF$. Therefore, the sides of $\triangle ABC$ are $16, 8, $ and $12$ units, and its perimeter is $16 + 8 + 12 = 36$ units.

3. **Justification for Option 2:**
   - The calculated perimeter of $\triangle ABC$ is 36 units, which matches Option 2. This confirms that Option 2 is the correct answer.

4. **Explanation of Incorrect Options:**
   - **Option 1 (18 units):** This is the perimeter of $\triangle DEF$, not $\triangle ABC$.
   - **Option 3 (48 units):** This value does not correspond to any logical calculation based on the given information.
   - **Option 4 (Cannot be determined due to insufficient data):** The data provided is sufficient to determine the perimeter of $\triangle ABC$ using the properties of medial triangles.

In conclusion, the correct answer is Option 2, as the perimeter of $\triangle ABC$ is 36 units.