In the figure above, M, N, and P are midpoints of the sides of an equilateral triangle whose perimeter is 18. What is the perimeter of the shaded region?
Show Hint
A triangle formed by joining the midpoints of the sides of a larger triangle will always have a perimeter that is exactly half the perimeter of the larger triangle. You could solve this problem instantly by calculating \(18 \div 2 = 9\).
Step 1: Understanding the Concept:
This problem utilizes the properties of an equilateral triangle and the Midpoint Theorem. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. Step 2: Detailed Explanation:
1. The outer triangle is equilateral and has a perimeter of 18.
The length of each side of the outer triangle is the perimeter divided by 3.
\[
\text{Side length of outer triangle} = \frac{18}{3} = 6
\]
2. M, N, and P are the midpoints of the sides of this triangle. The shaded region is the triangle \(\triangle MNP\) formed by connecting these midpoints.
3. According to the Midpoint Theorem, the length of each side of the inner triangle \(\triangle MNP\) is half the length of the corresponding parallel side of the outer triangle.
For example, the side MN is parallel to the base of the large triangle and its length is:
\[
\text{Length of MN} = \frac{1}{2} \times (\text{Side length of outer triangle}) = \frac{1}{2} \times 6 = 3
\]
4. Since the outer triangle is equilateral, all its sides are 6. Therefore, all the sides of the inner triangle will be half of 6.
\[
MN = NP = PM = 3
\]
This means the inner shaded triangle is also an equilateral triangle.
5. The perimeter of the shaded region (\(\triangle MNP\)) is the sum of the lengths of its sides.
\[
\text{Perimeter of } \triangle MNP = 3 + 3 + 3 = 9
\]
Step 3: Final Answer:
The perimeter of the shaded region is 9.
