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Congruence Transformations and Coordinate Proofs

Explore the three types of congruence transformations and learn how to demonstrate geometric properties using the Cartesian plane. Discover the properties of midsegments and their connection to triangle congruence. Practice coordinate proofs to prove geometric concepts.

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Congruence Transformations and Coordinate Proofs

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  1. Chapter 5 Intro to Congruence Transformations and Coordinate Proofs

  2. Objectives: • Identify the three types of congruence transformations. • Use the Cartesian plane to demonstrate Geometric properties

  3. Introduction Triangles are typically thought of as simplistic shapes constructed of three angles and three segments. As we continue to explore this shape, we discover there are many more properties and qualities than we may have first imagined. Each property and quality, such as the mid-segment of a triangle, acts as a tool for solving problems.

  4. Key Concepts The midpoint is the point on a line segment that divides the segment into two equal parts. A midsegment of a triangle is a line segment that joins the midpoints of two sides of a triangle.

  5. Key Concepts, continued In the diagram below, the midpoint of is X. The midpoint of is Y. A midsegment of is .

  6. Key Concepts, continued The midsegment of a triangle is parallel to the third side of the triangle and is half as long as the third side. This is known as the Triangle Midsegment Theorem.

  7. Key Concepts, continued Every triangle has three midsegments.

  8. Guided Practice: Example 1 Find the lengths of BC and YZ and the measure of ∠AXZ.

  9. Transformations • A transformation is an operation that maps an original geometric figure (the preimage) onto a new figure (the image). • Transformations are noted with an arrow. • Example: indicates that A is mapped to X, B is mapped to Y, and C is mapped to Z. • If the preimage and image are congruent figures, then the transformation is called a congruence (or rigid) transformation, or an isometry (iso – “same”)

  10. Congruence Transformations • There are 3 types of congruence transformations. (p. 294) • A reflection (or flip) is a transformation over a line called the line of reflection. Each point of the preimage and its image are the same distance away from the line of reflection. • A translation (or slide) is a transformation that moves all points of the preimage the same distance in the same direction. • A rotation (or turn) is a transformation around a fixed point called the center of rotation, through a specific angle and in a specific direction. Each point of the preimage and its image are the same distance from the center.

  11. Coordinate Proofs • The invention of the Cartesian (or coordinate) plane created connections between Algebra and Geometry. This allowed new discoveries and new techniques for examining geometric figures. • A coordinate proof uses figures in the coordinate plane and algebra to prove geometric concepts.

  12. Coordinate Proofs • To perform a coordinate proof, you must first position a figure (e.g. a triangle) on the coordinate plane: • Use the origin as a vertex or center of the triangle. • Place at least one side of the figure on an axis. • Keep the figure within the first quadrant if possible. • Use coordinates that make computations as simple as possible.

  13. Draw the midsegment connecting sides KL and JL. How can we do this? Use the midpoint formula. Can we make any observations about the relationship between the midsegment MN and the side JK? (Hint: check their slopes and lengths.) Observations about midsegments K (4, 5) 4 J (-2, 3) 2 N M 5 L (6, -1) -2 -4

  14. 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 Midsegment Theorem

  15. UW and VW are midsegments of ∆RST. Find UW and RT. UW = ½(RS) = ½ (12) = 6 RT = 2(VW) = 2(8) = 16 What would RU and UT be? RV and VS? Using the Midsegment Theorem R U 8 12 V T W S

  16. What would happen if you drew all three midsegments? How do the lengths of the midsegments compare to the other lengths? So what have we formed? Four congruent triangles (SSS). How do they relate to the original triangle? They are all similar to the original – each side is scaled by a factor of ½. (More on this later.) Connect all midpoints R U V T W S

  17. Guided Practice: The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle. 1. Plot the midpoints on a coordinate plane.

  18. Guided Practice: continued Connect the midpoints to form the midsegments , , and .

  19. Guided Practice: continued Calculate the slope of each midsegment. Calculate the slope of . The slope of is

  20. Guided Practice: continued Calculate the slope of . The slope of is 7.

  21. Guided Practice: continued Calculate the slope of . The slope of is .

  22. Guided Practice: Draw the lines that contain the midpoints. The endpoints of each midsegment are the midpoints of the larger triangle. Each midsegment is also parallel to the opposite side.

  23. Guided Practice: continued The slope of is . From point Y, draw a line that has a slope of .

  24. Guided Practice:continued The slope of is 7. From point X, draw a line that has a slope of 7.

  25. Guided Practice: continued The slope of is . From point Z, draw a line that has a slope of The intersections of the lines form the vertices of the triangle.

  26. Your turn Determine the vertices of the triangle. The vertices of the triangle are (–3, –2), (9, 4), and (–1, 12), as shown on the following slide.

  27. This is all for today!!

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