5.4 Midsegment Theorem

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5.4 Midsegment Theorem

• Geometry
• Mrs. Spitz
• Fall 2004

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Objectives

• Identify the midsegments of a triangle.
• Use properties of midsegments of a triangle.

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Assignment

• Pgs. 290-291 1-18, 21-22, 26-29

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Using Midsegments of a Triangle

• In lessons 5.2 and 5.3, you studied four special
  types of segments of a triangle
• Perpendicular bisectors
• Angle bisectors
• Medians and
• Altitudes
• A midsegment of a triangle is a segment that
  connects the midpoints of two sides of a
  triangle.

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How?

• You can form the three midsegments of a triangle
  by tracing the triangle on paper, cutting it out,
  and folding it as shown.
• Fold one vertex onto another to find one
  midpoint.
• Repeat the process to find the other two
  midpoints.
• Fold a segment that contains two of the
  midpoints.
• Fold the remaining two midsegments of the
  triangle.

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Ex. 1 Using midsegments

• Show that the midsegment MN is parallel to side
  JK and is half as long.
• Use the midpoint formula.

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Solution

• M -26 , 3(-1)
• 2 2
• M (2, 1)
• And
• N 46 , 5(-1)
• 2 2
• N (5, 2)
• Next find the slopes of JK and MN.

• Slope of JK 5 3 2 1
• 4-(-2) 6 3
• Slope of MN 2 1 1
• 5 2 3
• ?Because their slopes are equal, JK and MN are
  parallel. You can use the Distance Formula to
  show that MN v10 and JK v40 2v10. So MN is
  half as long as JK.

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Theorem 5.9 Midsegment Theorem

• The segment connecting the midpoints of two sides
  of a triangle is parallel to the third side and
  is half as long.
• DE AB, and
• DE ½ AB

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Ex. 2 Using the Midsegment Theorem

• UW and VW are midsegments of ?RST. Find UW and
  RT.
• SOLUTION
• UW ½(RS) ½ (12) 6
• RT 2(VW) 2(8) 16
• A coordinate proof of Theorem 5.9 for one
  midsegment of a triangle is given on the next
  slide. Exercises 23-25 ask for proofs about the
  other two midsegments. To set up a coordinate
  proof, remember to place the figure in a
  convenient location.

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Ex. 3 Proving Theorem 5.9

• Write a coordinate proof of the Midsegment
  Theorem.
• Place points A, B, and C in convenient locations
  in a coordinate plane, as shown. Use the
  Midpoint formula to find the coordinate of
  midpoints D and E.

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Ex. 3 Proving Theorem 5.9

• D 2a 0 , 2b 0 a, b
• 2 2
• E 2a 2c , 2b 0 ac, b
• 2 2
• Find the slope of midsegment DE. Points D and E
  have the same y-coordinates, so the slope of DE
  is 0.
• ?AB also has a slope of 0, so the slopes are
  equal and DE and AB are parallel.

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Now what?

• Calculate the lengths of DE and AB. The segments
  are both horizontal, so their lengths are given
  by the absolute values of the differences of
  their x-coordinates.
• AB 2c 0 2c DE a c a c
• ?The length of DE is half the length of AB.

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Objective 2 Using properties of Midsegments

• Suppose you are given only the three midpoints of
  the sides of a triangle. Is it possible to draw
  the original triangle? Example 4 shows one
  method.

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What?
PLOT the midpoints on the coordinate plane.
CONNECT these midpoints to form the midsegments
LN, MN, and ML. FIND the slopes of the
midsegments. Use the slope formula as shown.
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What?
ML 3-2 -1 2-4 2 MN 4-3 1
5-2 3 LN 4-2 2 2 5-4 1
Each midsegment contains two of the unknown
triangles midpoints and is parallel to the side
that contains the third midpoint. So you know a
point on each side of the triangle and the slope
of each side.
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What?
?DRAW the lines that contain the three
sides. The lines intersect at 3 different
points. A (3, 5) B (7, 3) C (1, 1) The perimeter
formed by the three midsegments of a triangle is
half the perimeter of the original triangle shown
in example 5.
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Ex. 5 Perimeter of Midsegment Triangle

• DF ½ AB ½ (10) 5
• EF ½ AC ½ (10) 5
• ED ½ BC½ (14.2) 7.1
• ?The perimeter of ?DEF is 5 5 7.1, or 17.1.
  The perimeter of ?ABC is 10 10 14.2, or 34.2,
  so the perimeter of the triangle formed by the
  midsegments is half the perimeter of the original
  triangle.