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Published byJack Strickland Modified over 8 years ago
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Coordinates and Design
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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
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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
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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
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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
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Plotting Points Examples Plot these points: A) (2, 5) B) (-2, 4) C) (4, -3) D) (-2, -5) E) (2, 0) F) (0, -4)
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Identifying Coordinates Examples A) B) C) D) E) F)
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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)
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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
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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)
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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
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Identify the Vertices The vertices of the shape are:
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1.3 - Transformations Transformations are movements of a geometric figure on a Cartesian plane These transformations can be translations, reflections and rotations
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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’)
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Example – Translation of a Figure What is the translation that occurred here?
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Example – Sketching a Translation Sketch the position of the image after a translation of 3 right, 4 up
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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)
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Example - Reflection Sketch the reflection image
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Example - Reflection Sketch the resulting image if the x-axis is the line of reflection
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Rotations Rotations are turns about a fixed center of rotation The image may be rotated clockwise or counterclockwise Rotations use 90 o increments
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Example - Rotation Rotate the triangle 180 o clockwise around point P
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Example - Rotation Rotate triangle ABC 90 o counterclockw ise around point P
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Finding Center of Rotation and Angle of Rotation Mark the Center of Rotation and the Angle of Rotation
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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
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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
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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
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Example – Multiple Transformations Rectangle ABCD is reflected in the line shown and then translated 2 left, 4 down
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Example – Multiple Transformations Triangle TUV is rotated 90 o counterclockwise around point P, and then reflected in the y-axis
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Example – Multiple Transformations Triangle FGH is rotated 180 o clockwise around point P and then translated 2 up, 3 left
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