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13.5 Coordinates in Space

13.5 Coordinates in Space. By: Emily Schneider Lindsey Grisham. Mission . Graph a rectangular solid Use the D istance point and Midpoint Formulas in space. Translating solids Dilating solids . Graphing.

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13.5 Coordinates in Space

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  1. 13.5 Coordinates in Space By: EmilySchneider Lindsey Grisham

  2. Mission • Graph a rectangular solid • Use the Distance point and Midpoint Formulas in space. • Translating solids • Dilating solids

  3. Graphing In space, each point requires three coordinates. This is because space has three dimensions. The x-, y-, and z-axes are all perpendicular to each other. A point in space is represented by an ordered triple. z y x

  4. Facts about Space • X- represents the depth • Y- represents the width • Z- represents the height

  5. Graphing a Rectangular Prism • Plot the x-coordinate first. Draw a segment from the origin _ units in the ± direction. • To plot the y-coordinate, draw a segment _ units in the ± direction. • Next, to plot the z-coordinate draw a segment _ units in the ± direction. • Label the point • Draw a rectangular prism and label each vertex. z y x

  6. Example 1 • Graph a rectangular solid that contains point A(-4,2,4) and the origin as vertices.

  7. Example 1 z y x

  8. Example 1 ~ Answer

  9. Formulas Distance formula for space: _____________________________________ Midpoint Formula for space:

  10. Example 2 (Distance) Find the Distance between T(6, 0, 0) and Q(-2, 4, 2).

  11. Example 2~ Answer Distance= =√[6-(-2) 2 + (o-4) 2 + (0-2) 2 = √(64+ 16 + 4) Answer= √84 or 2√21

  12. Example 3(Midpoint) • Determine the coordinates of the midpoint M of T(6, 0, 0) and Q(-2, 4, 2)

  13. Example 3~ Answer M of = = = (2, 2, 1)

  14. Translations In chapter 9 we learned how to translate a 2 dimensional shape. The same concept applies for translating a 3 dimensional shape. However, we have another coordinate (z) that we need to translate. First, write all of the vertices of the preimage in a chart. Next, add the ‘scale factor’ to the axis it specifies.

  15. Example 4 Find the coordinates of the vertices of the solid after the following translation. (x, y, z+20)

  16. Example 4~ answer

  17. Dilation using Matrices In chapter 9 we used a matrix to find the coordinates of a dilated image. The same concept works in space. First, write a matrix for the vertexes of the rectangular prism. Then, multiply the whole matrix by the scale factor.

  18. Example 5 • Dilate the prism to the left by a scale factor of 2. Graph the image after the dilation.

  19. Example 5 First, write a matrix for the vertexes of the rectangular prism. Then, multiply the whole matrix by the scale factor. Dilate these coordinates with a scale factor of 2. Original coordinates

  20. Example 5 ~ answer Original coordinates Translated coordinates Scale factor

  21. Example 5 • Now, we have the vertices of the dilated image. • The right is the dilated image graphed.

  22. Assignment Page 717 #10-15, 16-20 evens, 23-26, 35

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