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Chapter 1

Chapter 1. Coordinates and Design. What You Will Learn:. To use ordered pairs to plot points on a Cartesian plane To draw designs on a Cartesian plane To identify coordinates of the vertices of 2-D shapes To translate, reflect, and rotate points and shapes on a Cartesian plane

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Chapter 1

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  1. Chapter 1 Coordinates and Design

  2. What You Will Learn: • To use ordered pairs to plot points on a Cartesian plane • To draw designs on a Cartesian plane • To identify coordinates of the vertices of 2-D shapes • To translate, reflect, and rotate points and shapes on a Cartesian plane • To determine the horizontal and vertical distances between points

  3. 1.1 – The Cartesian Plane • A 17th century French mathematician (René Descartes) developed a system for graphing points • This system is known as a Cartesian plane • A Cartesian plane is also known as a coordinate grid

  4. A Cartesian Plane • This is a Cartesian plane • It has an x-axis, a y-axis, and an origin • The Cartesian plane is divided into 4 quadrants by the x and y axis

  5. Plotting Points • Each point placed on the Cartesian plane consists of an ordered pair • This ordered pair represents the x and y axis coordinates of the point • For example, the point (3, 5) has an x-coordinate value of +3, and a y-coordinate value of +5

  6. Plotting Points Examples • Plot these points: • (2, 5) • (-2, 4) • (4, -3) • (-2, -5) • (2, 0) • (0, -4)

  7. Identifying Coordinates Examples A) B) C) D) E) F)

  8. Some Helpful Hints • Always remember that ordered pairs come in the form (x, y) • Also remember that each point is measured from the origin (0, 0)

  9. 1.2 – Create Designs • Cartesian planes can be used to create designs • These designs are created by linking together a number of individual points on the Cartesian plane

  10. Create a Design • Plot the following points and connect them: E (2, 5) F (2, 2) G (5, 2) H (5, -2) I (2, -2) J (2, -5) K (-2, -5) L (-2, -2) M (-5, -2) N (-5, 2) P (-2, 2) Q (-2, 5)

  11. Vertices • The vertices of a shape are the points where two sides of a figure meet • Each vertex of a shape on a Cartesian plane should be represented by a different letter

  12. Identify the Vertices • The vertices of the shape are:

  13. 1.3 - Transformations • Transformations are movements of a geometric figure on a Cartesian plane • These transformations can be translations, reflections and rotations

  14. Translations • A translation is a movement of an entire figure up, down, left, or right on the Cartesian plane • During a translation, the object does not change the direction it faces – it only changes places • The new figure’s vertices are indicated using “prime” notation (for example, A is translated to A’)

  15. Example – Translation of a Figure • What is the translation that occurred here?

  16. Example – Sketching a Translation • Sketch the position of the image after a translation of 3 right, 4 up

  17. Reflections • A reflection produces a mirror image of the object • Each point is reflected in a mirror line • The new points should be the same distance from the mirror line as the original points (but in the opposite direction)

  18. Example - Reflection • Sketch the reflection image

  19. Example - Reflection • Sketch the resulting image if the x-axis is the line of reflection

  20. Rotations • Rotations are turns about a fixed center of rotation • The image may be rotated clockwise or counterclockwise • Rotations use 90o increments

  21. Example - Rotation • Rotate the triangle 180o clockwise around point P

  22. Example - Rotation • Rotate triangle ABC 90o counterclockwise around point P

  23. Finding Center of Rotation and Angle of Rotation • Mark the Center of Rotation and the Angle of Rotation

  24. 1.4 – Horizontal and Vertical Distances • Horizontal and vertical distances can be easily measured on a Cartesian plane • You simply need to count the number of squares horizontally and vertically between the two points

  25. Example – Determining Distances • What are the horizontal and vertical distances from Z to each of the points on the plane? A B C D E F

  26. Multiple Transformations • Often transformations can be combined to produce a new image • For instance, an object may be rotated and then translated to produce a new image

  27. Example – Multiple Transformations • Rectangle ABCD is reflected in the line shown and then translated 2 left, 4 down

  28. Example – Multiple Transformations • Triangle TUV is rotated 90o counterclockwise around point P, and then reflected in the y-axis

  29. Example – Multiple Transformations • Triangle FGH is rotated 180o clockwise around point P and then translated 2 up, 3 left

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