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The coordinate geometry that you first worked with in Session 6 is useful for describing transformations. Before looking at how that works, here are a few problems with coordinates to get you warmed up.
Do each part on a new set of axes.
a. | Find three different points with x-coordinate 0. Where are all the points with x-coordinate 0? |
b. | On a new set of axes, plot the point (1,1). Then draw a horizontal line through that point. Name three other points on that line. What do all the points on that line have in common? |
c. | Suppose v is the vertical line passing the point (3,7). Find the coordinates of three points that are on the line v and three points that are not on the line v. How can you tell if a point is on line v just by looking at the coordinates? |
d. | What are the coordinates of the intersection of the horizontal line through (5,2) and the vertical line through (-4,3)? |
e. | Name and plot five points whose first coordinate is the same as the second coordinate. Where are all such points? |
In this picture, k and m are horizontal lines.
a. | Find the coordinates of six points between the lines k and m. |
b. | Find the coordinates of six points that are not between the lines k and m. |
c. | How can you tell if a point is between the lines k and m by looking at its coordinates? |
Problem H3
Copy and complete the table below:
a. | On a piece of graph paper, plot the three points in column A. Connect them to form a triangle. Plot the three points in column B. Connect them to form a triangle. Describe how the two triangles are related. |
b. | On a new piece of graph paper, plot triangles A and C. Describe how they’re related. |
c. | Repeat the process for triangles A and D. |
d. | Repeat the same for the rest of the table. |
Start with any point (x,y). Reflect that point over the x-axis. What are the coordinates of the new point? How do they relate to the coordinates of the original point? Explain. (You may want to try several cases.)
Start with any point (x,y). Reflect that point over the y-axis. What are the coordinates of the new point? How do they relate to the coordinates of the original point? Explain. (You may want to try several cases.)
Start with any point (x,y). Reflect that point over the line y = x. What are the coordinates of the new point? How do they relate to the coordinates of the original point? Explain. (You may want to try several cases.)
Crowe, D. and Thompson, Thomas M. (Reston, VA: National Council of Teachers of Mathematics, 1987). Transformation Geometry and Archaeology. Learning and Teaching Geometry, K-12, 1987 Yearbook. Edited by Mary Montgomery Lindquist.
Reproduced with permission from the publisher. Copyright © 1987 by the National Council of Teachers of Mathematics. All rights reserved.
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Transformation Geometry and Archaeology
Problem H1
a. | Any three points on the vertical axis will do — for instance, (0,-3), (0,1), and (0,13). All the points with x-coordinate 0 are on the y-axis. |
b. | Any three points with the y-coordinate of 1 will do — for instance, (-7,1), (0,1), and (12,1). All the points on this horizontal line have y-coordinate 1. |
c. | Any three points with x-coordinate 3 will do — for instance, (3,-4), (3,0), and (3,11). A point is on the line v if its x-coordinate is 3. If its x-coordinate is anything other than 3, the point is not on the line. |
d. | The coordinates are (-4,2). |
e. | For instance, (-3,-3), (-2,-2), (0,0), (4,4), (15,15). All of these points are on the line y = x. |
Problem H2
a. | There are infinitely many points between the two lines — for instance, (-32,-1), (-17,1) (0,0), (33,3), (155,3.5), (1000,3.9). |
b. | There are infinitely many points which are not between the two lines — for instance, (-32,7), (-17,-15), (0,5), (33,7), (155,4.5), (1000,7.9). |
c. | A point is between the two lines if its y-coordinate is greater than -2 and less than 4. |
Problem H3
A |
B |
C |
D |
E |
F |
G |
(x,y) | (x + 3, y – 2) |
(-x,y) | (2x,2y) | (x – 1, y + 2) |
(y,-x) |
(-y,x) |
(2,1) |
(5,-1) |
(-2,1) |
(4,2) |
(1,3) |
(1,-2) |
(-1,2) |
(-4,0) | (-1,-2) |
(4,0) |
(-8,0) |
(-5,2) |
(0,4) |
(0,-4) |
(-5,4) |
(-2,2) |
(5,4) |
(-10,8) |
(-6,6) |
(4,5) |
(-4,-5) |
a. | Triangle B is the translation of triangle A 3 units to the right and 2 units down: |
b. | Triangle C is obtained by reflecting triangle A about the vertical axis: |
c. | Triangle D is obtained by stretching triangle A in both x- and y-direction by a factor of 2: |
d. | Triangle E is obtained by shifting triangle A 1 unit to the left and 2 units up: |
e. | Triangle F is obtained by reflecting triangle A about the vertical axis and then about the line y = x. Alternatively, it can also be obtained as a -90° rotation of triangle A about the origin (0,0). |
f. | Triangle G is obtained by reflecting triangle A about the horizontal axis and then about the line y = x. Alternatively, it can also be obtained as a 90° rotation of triangle A about the origin (0,0). |
For example, (-2,3) becomes (-2,-3). In general, reflecting (x,y) about the horizontal axis yields (x,-y). In other words, the x-coordinate is unchanged while the y-coordinate is the negative of the original y-coordinate.
Problem H5
For example, (-2,3) becomes (2,3). In general, reflecting (x,y) about the vertical axis yields (-x,y). In other words, the y-coordinate is unchanged while the x-coordinate is the negative of the original x-coordinate.
For example (-2,3) becomes (3,-2). In general, if a point (x,y) is reflected about the line y = x, its new coordinates are (y,x). In other words, what used to be the x-coordinate becomes the y-coordinate, and what used to be the y-coordinate becomes the x-coordinate.