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Geometry Day 31

Geometry Day 31. Intro to Congruence Transformations and Coordinate Proofs. Objectives:. Identify the three types of congruence transformations. Use the Cartesian plane to demonstrate Geometric properties. Transformations.

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Geometry Day 31

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  1. Geometry Day 31 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. 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”)

  4. 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.

  5. 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.

  6. 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. • Let’s look at the examples on pp. 301-302.

  7. The Midsegment of a Triangle • A midsegment of a triangle is a segment that connects the midpoints of two of its sides.

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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. A coordinate proof of the Midsegment Theorem C (2a, 2b) D E A (0, 0) B ( 2c, 0)

  13. 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. A coordinate proof of the Midsegment Theorem C (2a, 2b) D E A (0, 0) B ( 2c, 0)

  14. Calculate the lengths of DE and AB. The segments are both horizontal, so their lengths are given by 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. A coordinate proof of the Midsegment Theorem C (2a, 2b) D E A (0, 0) B ( 2c, 0)

  15. Coordinate Proofs • Coordinate proofs can be useful because they allow us to prove concepts that we might not otherwise have the tools to do so.

  16. Homework 20 • Workbook, p. 55-57 • Handout (Transformations in the Coordinate Plane)

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