Transformations in the Coordinate Plane
Transformations in the coordinate plane are often represented by "coordinate rules" of the form (x, y) --> (x', y'). This means a point whose coordinates are (x, y) gets mapped to another point whose coordinates are (x', y').
When possible, simple formulas are given for x' and y' in terms of x and y. For example, (x, y) --> (x + y, x – y) is a coordinate rule for some transformation and maps the points (0, 0), (2, 0), (2, 5), and (0, 5) as follows:
(0, 0) --> (0, 0)
(2, 0) --> (2, 2)
(2, 5) --> (7, –3)
(0, 5) --> (5, –5)
This transformation is not an isometry (it changes the size of any figure) and the image of the blue rectangle with those vertices is the red rectangle:
Translations of geometric figures in the coordinate plane can be determined by translating the x- and y-coordinates of points. Horizontal and vertical translations are the easiest. All other translations can be thought of as a composition of horizontal and vertical translations. The following examples illustrate this.
Example 1: Give a coordinate rule for translating a figure horizontally by 3 units.
Solution: A horizontal translation just changes the x-coordinates of all points, so the rule is (x, y) à (x + 3, y). To illustrate, the blue rectangle with vertices (0, 0), (2, 0), (2, 5), and (0, 5) is translated to the red rectangle with coordinates (3, 0), (5, 0), (5, 5), and (3, 5):
Example 2: Give a coordinate rule for a translation by a distance of 4 units at 30o.
Solution: Consider a point with coordinates (x, y) and its image with coordinates (p, q)
Draw a right triangle with the point and its image as the endpoints of the hypotenuse. This is a 30-60-90 triangle, so the side opposite the 30o angle is half the hypotenuse and the other side is that times the square root of 3. Therefore we have the following picture:
From this picture we see that
and q = x + 2
Therefore the coordinate rule is: (x, y) -->
When a point is reflected in a line, the line is the perpendicular bisector of the segment joining the point and its image. We will only consider coordinate rules for reflections in horizontal and vertical lines, and in the lines y = x and y = –x since the rules for lines in general involve messy details beyond the scope of this course.
Example 3: Give a coordinate rule for reflecting in the line vertical line x = 3.
Solution: Consider a point (x, y) and its image (p, q):
The y-coordinate of the image is the same as the y-coordinate of the preimage, so q = y. Since the line x = 3 bisects the segment from the point to its image, the horizontal distances from the point to the line and its image to the line are equal, so
3 – x = p – 3
Adding 3 to both sides tells us that p = 6 – x. Therefore the coordinate rule is:
(x, y) --> (6 – x, y)
Example 4: Give a coordinate rule for reflecting in the line y = x.
Solution: Again let the point and its image have coordinates (x, y) and (p, q), respectively. The line y = x is a 45o line through the origin, and the relation between the point and its image looks like this:
If we draw horizontal and vertical segments from the axes through the points and to the line y = x, we have the following:
Since the green line is at 45o, we can focus on two squares to see that q = x and p = y:
Thus, the coordinate rule is: (x, y) --> (y, x)
That is, when reflected in the line y = x, the coordinates of any point are transposed.
Coordinate Rules for Reflections
In general, the following coordinate rules for reflections can easily be established:
Reflection in x-axis: (x, y) --> (x, –y)
Reflection in y-axis: (x, y) --> (–x, y)
Reflection in y = x: (x, y) --> (y, x)
Reflection in y = –x: (x, y) --> (–y, –x)
We will only consider rotations about the origin of multiples of 90o.
Example 5: Give a coordinate rule for a rotation about the origin of 90o (counterclockwise).
Solution: Such a rotation is equivalent to reflections in two lines that intersect at the origin and are 45o apart. We could use the x-axis as the first line and the line y = x as the second. The composite of these reflections is:
(x, y) --> (x, –y) --> (–y, x)
That is, a rotation about the origin of 90o has the coordinate rule: (x, y) --> (–y, x)
Coordinate Rules for Rotations
In general, we can state the following coordinate rules for (counterclockwise) rotations about the origin:
For a rotation of 90o: (x, y) --> (–y, x)
For a rotation of 180o: (x, y) --> (–x, –y)
For a rotation of 270o: (x, y) --> (y, –x)
Dilations in the Coordinate Plane
First consider dilations with the origin as center. Then the coordinate rule for a dilation with scale factor k is simply this:
(x, y) --> (kx, ky).
Example 6: Triangle ABC has coordinates A(–1, –3), B(1, 1) and C(2, –3). Triangle DEF has coordinates D(2, 6), E(–4, 6) and F(–2, –2). Show that triangle DFE is the image of triangle ABC under a dilation with center at the origin, and find the scale factor.
Solution: The image of A is given by (–1, –3) --> (–1k, –3k). If D is that image, then –1k = 2 and –3k = 6. Both give k = –2. If we apply this dilation to B and C, we find that F is the image of B and E is the image of C.
Dilations with Center other than the Origin
A dilation with any point other than the origin as the center of dilation can be accomplished by first translating the center of dilation and figure so the origin becomes the center, and then translating back:
Example 7: Find a coordinate rule for the dilation with center (5, –3) and scale factor 2.
Solution: If (x, y) is a point on a figure to be dilated, we first translate left 5 and up 3. This gives us the point (x – 5, y + 3), and the origin becomes the center of the dilation. The dilation now gives us (2x – 10, 2y + 6). Then we translate back--that is, right 5 and down 3, which gives us (2x – 10 + 5, 2y + 6 – 3). So the coordinate rule is:
(x, y) --> (2x – 5, 2y + 3)
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