CC Geometry H Aim #12: How do we do transformations without the use of a coordinate plane?
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1 CC Geometry H im #12: How do we do transformations without the use of a coordinate plane? y o Now: Plot ΔBC with (3,2), B(3,6), and C(6,2) a) Reflect ΔBC over the x axis (r x-axis ) State the coordinates of the new triangle. ' ( ) B'( ) C'( ) b) Rotate ΔBC 90 o counterclockwise around the origin (R 0, ) 90 o State the coordinates of the new triangle. " ( ) B"( ) C"( ) x transformation is a change in the location or size of a figure. transformation,, of a plane is a function that assigns to each point P, the input, in the plane a unique point (P), the output, in the plane. We call a figure that is about to undergo a transformation the while the figure that has undergone the transformation is called the. Name the type of transformation shown: Which of the above transformations keep side lengths and angle measures the same? Transformations that preserve lengths of segments and measures of angles are called basic motions. They are: Why is a dilation not a rigid motion? Use the facts above to clearly explain your answer.
2 Rigid motions are also known as isometric transformations or isometries. The pre-image and image following a rigid transformation are identical (congruent). Link to read further about isometries: Rotation or a rotation, we need to know: a) center of rotation b) direction: Clockwise (CW) or Counterclockwise (CCW) c) number of degrees rotated. xample: Use your protractor to find the angle of rotation and name it 3 ways: Note: Pre-image: (solid line) Image: (dotted line) Center of rotation: Reflection or a reflection: The line of reflection acts as the perpendicular bisector of each segment that joins a given vertex of the pre-image with the respective vertex of the image. Translation or a translation, we need to know: a) the point to be translated b) the length and direction of the vector xample: a) By what rigid motion are ΔBC and Δ congruent? b) raw and label ΔGHI congruent to ΔBC by reflection across the x-axis. c) raw and label ΔJKL congruent to Δ by a translation 1 unit right and 4 units down, T 1,-4 B C
3 We will use the three rigid motions to find images. 1. Square BC is inscribed in circle (each vertex touches the circumference of circle ). ind the missing image. B H C G Pre-image Transformation Symbol Image point reflection over G r G () I point B reflection over point r (B) point reflection over G using the same translation that maps C onto B translation along vector I rg (BC) T? () T I (G) ΔHC reflection over G r G (ΔHC) point H point rotation 900 clockwise about center rotation 90 0 counterclockwise about center ΔB reflection over HI r HI (ΔB) ΔB rotation about center translation along vector H T H () point reflection over C r C ()
4 2. BC is a regular hexagon (equal sides and equal angles), inscribed in a circle, center G. B G C a) What is the image of point after a reflection over C? r C () = b) What is the image of the point B after a reflection over point G? r G (B) = c) What is the image of BC after a reflection over? r (BC) = d) What is the image of point after a reflection over the B? r B ()= e) What is the image of point B after a 60 o counterclockwise rotation around G? R G, (B) = 60 o f) What is the image of C after a 180 o counterclockwise rotation around G? (C) = R G, 180 o g) If the image of point after a translation is, what is the image of point under the same translation? h) What is the image of triangle G after a reflection over C? r C (ΔG) =
5 Geometry ssumptions We have now done some work with all three rigid motions (rotations, reflections, and translations). a. ny basic rigid motion preserves lines, rays, and segments. That is, for a basic rigid motion of the plane, the image of a line is a line, the image of a ray is a ray, and the image of a segment is a segment. b. ny basic rigid motion preserves lengths of segments and angle measures of angles. Let's Sum it Up!! ctivity: Use the three rigid transformations (isometries) in this activity to show shapes are congruent. Basic Rigid Motion: basic rigid motion is a rotation, reflection, or translation of the plane. Given a transformation, the image of a point is the point to which is mapped by the transformation. istance Preserving: transformation is said to be distance preserving if the distance between the images of two points is always equal to the distance between the pre-images of the two points. ngle Preserving: transformation is said to be angle preserving if (1) the image of any angle is again an angle and (2) for any given angle, the angle measure of the image of that angle is equal to the angle measure of the pre-image of that angle.
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