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 To determine the horizontal and vertical distances between points

1.1 – The Cartesian Plane A 17 th 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

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

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

Plotting Points Examples Plot these points: A) (2, 5) B) (-2, 4) C) (4, -3) D) (-2, -5) E) (2, 0) F) (0, -4)

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

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)

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

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)

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

Identify the Vertices The vertices of the shape are:

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

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’)

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

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

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)

Example - Reflection Sketch the reflection image

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

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

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

Example - Rotation Rotate triangle ABC 90 o counterclockw ise around point P

Finding Center of Rotation and Angle of Rotation Mark the Center of Rotation and the Angle of Rotation

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

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

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

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

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

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