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Midsegments of Triangles LESSON 5-1 Additional Examples In XYZ, M, N, and P are midpoints. The perimeter of MNP is 60. Find NP and YZ. Because the perimeter of NP + MN + MP = 60 NP + 24 + 22 = 60 NP + 46 = 60 NP = 14 MNP is 60, you can find NP. Definition of perimeter Substitute 24 for MN and 22 for MP. Simplify. Subtract 46 from each side. Use the Triangle Midsegment Theorem to find YZ. MP = 1 YZ Triangle Midsegment Theorem 2 22 = 1YZ 2 Substitute 22 for MP. 44 = YZ Multiply each side by 2. Quick Check HELP GEOMETRY Midsegments of Triangles LESSON 5-1 Additional Examples Find m AMN and m ANM. MN and BC are cut by transversal AB , so B are corresponding angles. AMN and MN || BC by the Triangle Midsegment Theorem, so AMN B because parallel lines cut by a transversal form congruent corresponding angles. m AMN = 75 because congruent angles have the same measure. In AMN, AM = AN, so m Triangle Theorem. m ANM = m ANM = 75 by substituting 75 for m AMN by the Isosceles AMN. Quick Check HELP GEOMETRY Midsegments of Triangles LESSON 5-1 Additional Examples Explain why Dean could use the Triangle Midsegment Theorem to measure the length of the lake. Solution: To find the length of the lake, Dean starts at the end of the lake and paces straight along that end of the lake. He counts the number of strides (35). Where the lake ends, he sets a stake. He paces the same number of strides (35) in the same direction and sets another stake. The first stake marks the midpoint of one side of a triangle. Dean paces from the second stake straight to the other end of the lake. He counts the number of his strides (236). HELP GEOMETRY Midsegments of Triangles LESSON 5-1 Additional Examples (continued) Dean finds the midsegment of the second side by pacing exactly half the number of strides back toward the second stake. He paces 118 strides. From this midpoint of the second side of the triangle, he returns to the midpoint of the first side, counting the number of strides (128). Dean has paced a triangle. He has also formed a midsegment of a triangle whose third side is the length of the lake. By the Triangle Midsegment Theorem, the segment connecting the two midpoints is half the distance across the lake. So, Dean multiplies the length of the midsegment by 2 to find the length of the lake. HELP Quick Check GEOMETRY