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Warm-Up

Warm-Up. A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments. 5.1 Midsegment Theorem and Coordinate Proof. Objectives: To discover and use the Midsegment Theorem To write a coordinate proof. Midsegment.

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Warm-Up

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  1. Warm-Up A midsegmentof a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

  2. 5.1 Midsegment Theorem and Coordinate Proof Objectives: • To discover and use the Midsegment Theorem • To write a coordinate proof

  3. Midsegment A midsegmentof a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

  4. Example 1 Graph ΔACE with coordinates A(-1, -1), C(3, 5), and E(7, -5). Graph the midsegment MS that connects the midpoints of AC and CE.

  5. Example 1 Now find the slope and length of MS and AE. What do you notice about the midsegment and the third side of the triangle? They are parallel

  6. Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

  7. Example 2 The diagram shows an illustration of a roof truss, where UV and VW are midsegments of ΔRST. Find UV and RS. RS = 114 UV = 45

  8. Example 3 1. 2. 7

  9. Coordinate Proof Coordinate proofs are easy. You just have to conveniently place your geometric figure in the coordinate plane and use variables to represent each vertex. • These variables, of course, can represent any and all cases. • When the shape is in the coordinate plane, it’s just a simple matter of using formulas for distance, slope, midpoints, etc.

  10. Example 4 Place a rectangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

  11. Example 4 Convenient placement usually involves using the origin as a vertex and lining up one or more sides of the shape on the x- or y-axis.

  12. Example 5 Place a triangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

  13. Example 6 Place the figure in the coordinate plane in a convenient way. Assign coordinates to each vertex. • Right triangle: leg lengths are 5 units and 3 units • Isosceles Right triangle: leg length is 10 units Do this in your notebook

  14. Example 7 A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

  15. Example 8 Find the missing coordinates. Then show that the statement is true. S(1/2h,1/2k) T(1 1/2h,1/2k) d=√(1.5h-0)2+(.5k-0)2 d=√2.25h + .25k d=√(.5h-2h)2+(.5k-0)2 d=√-1.5h+.25k

  16. Example 9 Write a coordinate proof for the Midsegment Theorem. Given: MS is a midsegment of ΔOWL Prove: MS || OL and MS = ½OL

  17. Example 10 Explain why the choice of variables below might be slightly more convenient. Given: MS is a midsegment of ΔOWL Prove: MS || OL and MS = ½OL M=(b,c) S=(b+a,c) Find slopes of MS & OL Find distance of MS and OL Work it out in your notebook

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