Slide 1 / 273 Slide 2 / 273 Geometry Transformations 2015-10-26 www.njctl.org Slide 3 / 273 Slide 4 / 273 Table of Contents Throughout this unit, the Standards for Mathematical Practice click on the topic to go to that section are used. Transformations MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Translations MP3: Construct viable arguments and critique the reasoning of others. Reflections MP4: Model with mathematics. MP5: Use appropriate tools strategically. Rotations MP6: Attend to precision. Identifying Symmetry with Transformations MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Composition of Transformations Additional questions are included on the slides using the "Math Congruence Transformations Practice" Pull-tabs (e.g. a blank one is shown to the right on Dilations this slide) with a reference to the standards used. Similarity Transformations If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. PARCC Sample Questions Slide 5 / 273 Slide 6 / 273 Transformational Geometry Transformations were referenced in the below slide from the first unit in Geometry. This was the definition Euclid used for congruence. We just used the word "moved" instead of "transformed". Two objects are congruent if they can be moved, by any Transformations combination of translation, rotation and reflection, so that every part of each object overlaps. This is the symbol for congruence: If a is congruent to b, this would be shown as below: a b ≅ Return to Table of Contents which is read as "a is congruent to b."
Slide 7 / 273 Slide 8 / 273 Transformations Transformational Geometry The reason for that is that although two objects could look the same, that doesn't prove that they are the same. Even though the idea that objects are congruent if they can be moved so that they match up is an underlying idea of geometry, it was not used directly by Euclid. We would never accept as a proof that two figures in a drawing look like they would line up, therefore they are congruent. Nor have we used that idea directly in this course, so far. However, with analytic geometry, it became possible to exactly "move" an object and prove if all its points lined up with a second Instead Euclid developed what we now call Synthetic Geometry, object. based on the idea that we can construct objects which are congruent. If so, the objects are congruent. That's what we've been using so far. This is transformational geometry. Slide 9 / 273 Slide 10 / 273 Transformations Transformations In 1872 Felix Klein created an approach to mathematics based on using transformations to prove objects congruent. First, let's review the ideas of rigid transformations and then we'll develop some new notation to keep those ideas straight. Transformational Geometry built on the synthesis of algebra and geometry Remember, rigid transformations move an object without changing its created by Analytic Geometry. size or shape. It led to a further synthesis with Group They include translations, rotations and reflections. Theory and other forms of mathematics which were being developed in more Dilations are NOT rigid transformations since they change the size of advanced algebra. the object, even though they preserve its shape. We will explore this using Felix Klein transformations on the Cartesian plane to prove that figures are congruent or 1849 - 1925 similar. Slide 11 / 273 Slide 12 / 273 Transformations Transformations In a transformation, the original figure is the preimage, and the A transformation of a geometric figure is a mapping that results in a resulting figure is the image. change in the position, shape, or size of the figure. In the examples below, the preimage is green and the image is pink. In the game of dominoes, you often move the dominoes by sliding them, turning them or flipping them. Each of these moves is a type of transformation. translation - slide reflection - flip rotation - turn
Slide 13 / 273 Slide 14 / 273 Transformations Transformations Which of these is a rigid motion? Some transformations (like the dominoes) preserve distance and angle measures. These transformations are called rigid motions. Rotation-turn Translation- slide To "preserve distance" means that the distance between any two points of the image is the same as the distance between the corresponding points of the preimage. Since distance is preserved in rigid motions, so will other measurements in the figures that rely on distance. For example, both the perimeter and area of the shape will remain the same after the Reflection- Flip Dilation - Size change completion of a rigid motion. To "preserve angles" means that the angles of the image have the same measures as the corresponding angles in the preimage. Slide 15 / 273 Slide 16 / 273 Transformations 1 Does the transformation appear to be a rigid motion? Explain. A transformation maps every point of a figure onto its image and may A Yes, it preserves the distance between consecutive be described using arrow notation ( ). points. Prime notation ( ' ) is sometimes used to identify image points. B No, it does not preserve the distance between consecutive points. In the diagram below, A' is the image of A . A' A Δ ABC Δ A'B'C' Δ ABC maps onto Δ A'B'C' B B' C C' Preimage Image Note: You list the corresponding points of the preimage and image in the same order, just as you would for corresponding points in congruent figures or similar figures. Slide 17 / 273 Slide 18 / 273 2 Does the transformation appear to be a rigid motion? 3 Which transformation is not a rigid motion? Explain. A Yes, distances are preserved. A Reflection B Yes, angle measures are preserved. B Translation C Both A and B. D No, distances are not preserved. C Rotation D Dilation Preimage Image
Slide 19 / 273 Slide 20 / 273 4 Which transformation is demonstrated? 5 Which transformation is demonstrated? A Reflection A Reflection B Translation B Translation C Rotation C Rotation D Dilation D Dilation Slide 21 / 273 Slide 22 / 273 7 Quadrilateral ABCD below is reflected about line m. After 6 Which transformation is demonstrated? the reflection, how is the perimeter of A'B'C'D' be related to the perimeter of ABCD? A Reflection B Translation C Rotation D Dilation m A Because reflection is a rigid motion that preserves the distance between consecutive points, the perimeter will remain the same. B Because reflection is not a rigid motion and does not preserve the distance between consecutive points, the perimeter will change. Slide 23 / 273 Slide 24 / 273 8 Quadrilateral ABCD below is rotated 180°. After the rotation, how is the area of A'B'C'D' be related to the area of ABCD? Translations A Because rotation is a rigid motion that preserves the distance between consecutive points, the area will remain the same. Return to Table of B Because rotation is not a rigid motion and does not Contents preserve the distance between consecutive points, the area will change.
Slide 25 / 273 Slide 26 / 273 Translations Translations A translation is a transformation that maps all points of a figure That means that any line drawn from a point on the preimage the same distance in the same direction. to the corresponding point on the image will be of equal length. B' B' AA' = BB' = CC' B B C' C' A' A' A C A C Slide 27 / 273 Slide 28 / 273 Translations Translations That also means that the lengths of the sides of the preimage And, that the corresponding angles in the preimage and image and image will be the same. are congruent. B' B' AB = A'B', m ∠ A = m ∠ A' B B m ∠ B = m ∠ B' BC = B'C', m ∠ C = m ∠ C' AC = A'C' A' C' C' A' A C A C Slide 29 / 273 Slide 30 / 273 Translations Translations If I can show that translating each of the vertices of ΔABC in the same way results in them lining up with all the vertices of So far, so good...but if I'm given these two triangles, how do I ΔA'B'C' then we have proven those Δs ≅ . prove them congruent based on using transformations. B' That's made possible by adding in Analytic Geometry through the use of a Cartesian plane. B B' A' C' B A C C' A' A C
Recommend
More recommend
