The perimeter of a triangle with midsegments (MN), (NO), and (OP) is the sum of the lengths of these midsegments.
In a triangle, midsegments connect the midpoints of its sides. If lines (MN), (NO), and (OP) are midsegments of a triangle, they divide the triangle into four smaller congruent triangles. These midsegments are parallel to the sides of the original triangle, creating smaller triangles that are similar to the original one.
Since each of these smaller triangles is similar to the original triangle, the ratio of their corresponding sides is the same. The midsegments divide each side of the original triangle in half.
If (AB), (BC), and (CA) are the sides of the original triangle, and (M), (N), and (O) are the midpoints of these sides, then (MN) is parallel to (BC), (NO) is parallel to (CA), and (OP) is parallel to (AB).
The perimeter of the original triangle is the sum of its sides, (AB + BC + CA). Since the midsegments divide each side in half, the perimeter of the original triangle is also the sum of the lengths of (MN), (NO), and (OP).
Therefore, the perimeter is (MN + NO + OP).