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13-5 Coordinates in Space

By Danny Nguyen and Jimmy Nguyen . 13-5 Coordinates in Space. Objectives. Graph solids in space. Use the Distance and Midpoint Formulas for points in space. Ordered Triples. In the coordinate plane we used an ordered pair with 2 real numbers to determine a point ( x,y )

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13-5 Coordinates in Space

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  1. By Danny Nguyen and Jimmy Nguyen 13-5 Coordinates in Space

  2. Objectives • Graph solids in space. • Use the Distance and Midpoint Formulas for points in space.

  3. Ordered Triples • In the coordinate plane we used an ordered pair with 2 real numbers to determine a point (x,y) • In space, we need 3 real numbers to graph a point. This is because space has 3 dimensions. These numbers make up an ordered triple (x,y,z).

  4. Space + • In space, the x-, y-, and z- axes are perpendicular to each other. • X represents the depth • Y represents the width • Z represents the height • Notice how P(2,3,6) is graphed. _ _ + + _

  5. Example 1: How to Graph a Shape in Space • Graph a rectangular solid that contains point A(-4,2,4) and the origin as vertices.

  6. Example 1:How to Graph a Shape in Space • Plot the x-coordinate first. Go 4 units in the negative direction. • Next, plot the y-coordinate. Go 2 units in the positive direction. • Finally, plot the z- coordinate. 4 units in the positive direction • We have now plotted coordinate A. • Draw the rest of the rectangular prism.

  7. Distance Formula in Space • Remember Distance Formula from the coordinate plane? We also have a formula for distance in Space.

  8. Proof of the Distance Formula in Space

  9. Example 2: Distance Formula • Find the Distance between T(6, 0, 0) and Q(-2, 4, 2).

  10. Your Turn : (Distance Formula) • Find the distance between A(3, 1, 4) and B(8, 2, 5) AB AB ( ) + ( ) + ( ) ( ) + ( ) + ( ) Answer: √27 3 3 OR

  11. Midpoint Formula in Space • We also have a formula for Midpoints in Space.

  12. Midpoint Formula Explanation • An average is defined as the middle measure of a data set. • When we use midpoint formula, we are basically finding the average between the x, y, and z, coordinates. • Putting the averages together to make an ordered triple lets us find where the midpoint of the segment is in space.

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

  14. Your Turn: (Midpoint Formula) • Find the coordinates of the midpoint M of AB. A(3, 1, 4) and B(8, 2, 5) = ( , , ) Answer: (Secant), just kidding :P it is (11/2, 3/2, 9/2) or (5.5, 1.5, 4.5)

  15. Translating a Solid • Remember Translations? You can also do translations in space with solids. • It is basically the same principal we saw in Ch. 9 except we have another coordinate to translate.

  16. Example 4: Translating a Solid • Find the coordinates of the vertices of the solid after the following translation. (x, y, z+20)

  17. Example 4: Translating a Solid

  18. Dilation with Matrices • We should also remember what a dilation is from Ch. 9. We used a matrix to find the coordinates of an image after a dilation. We can also do the same thing here.

  19. Example 5: Dilation with Matrices • Dilate the prism to the right by a scale factor of 2. Graph the image after the dilation.

  20. Example 5: Dilation with Matrices • First, write a vertex matrix for the rectangular prism. • Next, multiply each element of the vertex by the scale factor of 2.

  21. Example 5: Dilation with Matrices • We now have the vertices of the dilated image. • To the right we have a graph of the dilated image.

  22. Trololololololol… uhh Colby’s idea O.o • Your homework: • Pre-AP Geometry: Pg 717 #11, 12, 14, 15-26, 28, 30 • Have fun doing 16 problems! 

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