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Midsegments of Triangles 5.1

Learn to define and apply midsegment properties, understand coordinate proofs, and solve unknowns using Triangle Midsegment Theorem. Practice with examples and coordinate geometry.

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Midsegments of Triangles 5.1

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  1. Midsegments of Triangles5.1 Today’s goalsBy the end of class today, YOU should be able to… 1. Define and use the properties of midsegments to solve problems for unknowns. 2. Use the properties of midsegments to make statements about parallel segments in a given triangle. 3. Understand and write coordinate proofs.

  2. Midsegments • A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle.

  3. 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 half its length.

  4. Proving the Triangle Midsegment Theorem… • To prove the Triangle Midsegement Theorem, use coordinate geometry and algebra. This style of proof is called a coordinate proof. To begin the proof: • Place the triangle in a convenient spot on the coordinate plane. • Choose variables for the coordinates of the vertices.

  5. Coordinate proofs cont… • State what you are given. • State what you want to prove. • Use the midpoint formula to find the midpoints of two sides of the triangle. • To prove that the midsegment and third side are parallel, show that their slopes are equal. • To prove that the midsegment is half the length of the third side, use the distance formula to calculate the length of both segments. *See section 5.1 for a complete version of a coordinate proof for the Triangle Midsegment Theorem

  6. Ex.1: Find the lengths Using the following illustration (where H, J, and K are the midpoints), find the lengths of HJ, JK, and FG:

  7. Ex.1: Solution HJ = ½ EG = ½ (100) = 50 Use the Triangle Midsegment Theorem to show that the midsegment = ½ the length of the third side JK = ½ EF = ½ (60) = 30 HK = ½ FG FG = 2 HK = 2 (40) = 80

  8. You Try… Use the following triangle to solve the questions below (A, B, & C are midpoints): 1. DF=120, EF=90, BC=40. Find AB, AC, & DE. 2. EF=10x, AB=4x, AC=10, DE=12x. Find BC & AC.

  9. Ex.2: Identify the parallel segments If A, B, & C are midpoints, which segment is parallel to AC?

  10. Ex.2: Solution By the Triangle Midsegment Theorem, AC ll EF

  11. You Try… If A, B, & C are midpoints: • Which segment is parallel to DE? • Which segment is parallel to AB?

  12. Practice Solve with a partner of your choice… The triangular face of the Rock and Roll Hall of Fame in Cleveland, Ohio is isosceles. The length of the base is 229 ft 6 in. What is the length of a segment located half way up the face of the Rock and Roll Hall of Fame Explain your reasoning.

  13. Homework • Page 246 #s 2, 4, 6, 7, 13, 30 • Page 247 #s 26, 34, 35 • The assignment can also be found online at: • http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037878-X/Ch05/05-01/PH_Geom_ch05-01_Ex.pdf

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