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By Danny Nguyen and Jimmy Nguyen Graph solids in space. Use the Distance and Midpoint Formulas for points in space. 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). + 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. _ _ + + _ Graph a rectangular solid that contains point A(-4,2,4) and the origin as vertices. Plot the x-coordinate first. Go 4 units in the negative direction. Next, plot the ycoordinate. Go 2 units in the positive direction. Finally, plot the zcoordinate. 4 units in the positive direction We have now plotted coordinate A. Draw the rest of the rectangular prism. Remember Distance Formula from the coordinate plane? We also have a formula for distance in Space. Find the Distance between T(6, 0, 0) and Q(-2, 4, 2). Find the distance between A(3, 1, 4) and B(8, 2, 5) AB AB ( ( ) + ( ) + ( ) + ( ) + ( Answer: √27 OR ) ) 3 3 We also have a formula for Midpoints in Space. 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. Determine the coordinates of the midpoint M of . T(6, 0, 0) and Q(-2, 4, 2) 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) 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. Find the coordinates of the vertices of the solid after the following translation. (x, y, z+20) 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. Dilate the prism to the right by a scale factor of 2. Graph the image after the dilation. First, write a vertex matrix for the rectangular prism. Next, multiply each element of the vertex by the scale factor of 2. We now have the vertices of the dilated image. To the right we have a graph of the dilated image. Your homework: Pre-AP Geometry: Pg 717 #11, 12, 14, 15-26, 28, 30 Have fun doing 16 problems!