The sides of a triangle are 5 cm, 6 cm, and 7 cm. One more triangle is formed by joining the mid-points of the sides. The perimeter of the second triangle in cm is

### Concept & Midpoint Theorem for Perimeter According to the midpoint theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is exactly half of its length. Consequently, the perimeter of the inner triangle formed by joining the midpoints is exactly half the perimeter of the original triangle. $$ \text{Perimeter}_{\text{new}} = \frac{\text{Perimeter}_{\text{original}}}{2} $$ ### Step-by-Step Solution 1. Calculate the perimeter of the original triangle by summing its sides: $\text{Original Perimeter} = 5 + 6 + 7 = 18\text{ cm}$. 2. The sides of the new triangle are half the lengths of the original sides: New sides = $\frac{5}{2}$, $\frac{6}{2}$, and $\frac{7}{2}$ (which are $2.5\text{ cm}$, $3\text{ cm}$, and $3.5\text{ cm}$). 3. Calculate the perimeter of the new triangle: $\text{New Perimeter} = 2.5 + 3 + 3.5 = 9\text{ cm}$. (Alternatively, just divide the original perimeter by 2: $\frac{18}{2} = 9\text{ cm}$). ### Exam Strategy & Shortcut Save time by finding the sum of the original sides and immediately halving the result. $5 + 6 + 7 = 18 \rightarrow \frac{18}{2} = 9$. ### Common Pitfall A common mistake is trying to divide the area by 4 and then reverse-engineering the side lengths. Perimeter scales linearly (factor of $\frac{1}{2}$), while area scales quadratically (factor of $\frac{1}{4}$). ### Final Answer Therefore, the correct answer is **9**.