Do Investigation On page 243 Section 5-1 Triangle Midsegments SPI 32J: identify the appropriate segment of a triangle given a diagram and vs (median, altitude, angle and perpendicular bisector) Objectives: Use properties of midsegment to solve problems Do Investigation On page 243 Line segment LN is the midsegment of the triangle (connects the midpoint of the two sides) LN = ½ AB

Triangle Midpoint Theorem Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is ½ its length. Length of Midsegment = ½ length of base

Finding Lengths using Triangle Midpoint Theorem In ∆ XYZ, M, N, and P are midpoints. The perimeter of ∆ MNP is 60. Find NP and YZ. Because the perimeter of MNP is 60, you can find NP. NP + MN + MP = 60 Definition of perimeter NP + 24 + 22 = 60 Substitute 24 for MN and 22 for MP. NP + 46 = 60 Simplify. NP = 14 Subtract 46 from each side. Use the Triangle Midsegment Theorem to find YZ. MP = YZ Triangle Midsegment Theorem 22 = YZ Substitute 22 for MP. 44 = YZ Multiply each side by 2. 1 2

Apply Midpoint Theorem Find m AMN and m ANM. MN and BC are cut by transversal AB , so AMN and B are corresponding angles. 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 ANM = m AMN by the Isosceles Triangle Theorem. m ANM = 75 by substituting 75 for m AMN.

Real World: Apply Midpoint Theorem Indirect Measurement. Kate wants to paddle her canoe across the lake. To determine how far she must paddle, she paced out a triangle counting the number of strides as shown. a. If Kate’s strides average 3.5 ft, what is the length of the longest side of the triangle? b. What distance must Kate paddle across the lake? a. 1050 ft b. 437.5 ft