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 Investigati

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Line segment LN is the midsegment of the triangle(connects the midpoint of the two sides)

LN = ½ AB

Triangle Midpoint Theorem

Length of Midsegment = ½ length of base

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.

In ∆ XYZ, M, N, and P are midpoints.

The perimeter of ∆ MNP is 60. Find

NP and YZ.

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.

Because the perimeter of MNP is 60, you can find NP.

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.

1212

Finding Lengths using Triangle Midpoint Theorem

Find m AMN and m ANM.

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.

MN || BC by the Triangle Midsegment Theorem, so AMN B because parallel lines cut by a transversal form congruent corresponding angles.

MN and BC are cut by transversal AB , so AMN and B are corresponding angles.

Apply Midpoint Theorem

Real World: Apply Midpoint Theorem

b. 437.5 ft

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