Transformations in the coordinate plane
- 1. TRANSFORMATIONS IN THE COORDINATE PLANE Notes
- 2. IN THIS LESSON, WE WILL: • Review the four geometric transformations • Identify unique geometric properties of each geometric transformation • Determine which geometric transformations produce congruent figures • Identify which geometric transformations produce similar figures • Practice transforming figures in the coordinate plane • Use appropriate notation to represent all geometric transformations
- 3. TRANSLATION • Moves an object up, down, left, and/or right in the coordinate plane • Slides an object around • Preserves shape AND size • All distances are the same • Creates congruent figures
- 4. TRANSLATION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientation of the pre- image and the image are the same • A special property of translations is that the distances between corresponding vertices on the pre- image and the image are equal
- 5. REFLECTION • Flips an object across a line of reflection in the coordinate plane • Changes an object’s orientation (direction) • Preserves shape AND size • Creates congruent figures
- 6. REFLECTION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientations of the pre- image and the image are different • A special property of reflections is that the segment from any vertex on the pre-image to its corresponding vertex on the image is a perpendicular bisector of the line of reflection
- 7. ROTATION • Spins an object about a fixed point in the coordinate plane • Usually changes an object’s orientation (direction) • Preserves shape AND size • Creates congruent figures
- 8. ROTATION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientations of the pre- image and the image are different, unless you rotate one 360 degrees
- 9. A SPECIAL PROPERTY OF ROTATIONS • A point on the pre-image and its corresponding point on the image lie on a circle whose center is the center of rotation.
- 10. ANOTHER SPECIAL PROPERTY OF ROTATIONS • Line segments connecting corresponding points on the pre-image and image to the center of rotation are congruent and form an angle whose measure equals that of the angle of rotation.
- 11. DILATION • An enlargement or reduction of a figure by a scale factor in the coordinate plane • Changes the size of an object • Preserves shape • Creates similar figures
- 12. DILATION EXAMPLE • Notice that the pre-image and the image are similar: corresponding angles are congruent and corresponding sides are proportional • Notice that the orientations of the pre- image and the image are the same • A special property of dilated figures is that you can compute the scale factor of the dilation by finding the ratio 𝑁𝐸𝑊 𝑂𝐿𝐷
- 13. SUMMARY PRESERVES SHAPE • Translation • Reflection • Rotation • Dilation PRESERVES SIZE • Translation • Reflection • Rotation
- 14. SUMMARY PRESERVES ORIENTATION • Translation • Dilation
- 15. CONGRUENT VS. SIMILAR FIGURES PRODUCES CONGRUENT FIGURES • Translation • Reflection • Rotation PRODUCES SIMILAR FIGURES • Dilation
- 16. PRACTICE USING TRANSFORMATIONS AND THEIR PROPERTIES Work through this example in your notes.
- 17. 1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Notice the coordinates: • A(2, 2) • B(-2, 3) • C(-5, 0) • We will do the translation first. We have to move the triangle left three units and up two units. This means f(x, y) = (x – 3, y + 2).
- 18. 1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Notice the new coordinates after we subtract 3 from each x and add 2 to each y: • A(-1, 4) • B(-5, 5) • C(-8, 2) • Next, let’s dilate the triangle by a factor of ½. This would be represented by f(x, y) = (1/2x, 1/2y).
- 19. 1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Now we have our new triangle: • A’ (-1/2, 2) • B’ (-5/2, 5/2) • C’ (-4, 1) • We put the two changes together to write a function rule that takes us directly from the pre-image to the image. This would be represented by f(x, y) = (1/2(x - 3), 1/2(y + 2), or f(x, y) = (1/2x – 3/2, 1/2y + 1)