Concepts and Connections

Shape

Transformation Geometry--Concepts & Connections

Directed Activity Translation. Coordinate geometry provides a number of tools for studying and applying transformations. In traditional geometry, a translation is defined using a drawn slide arrow like BB” in Figure 5.24. In coordinate geometry, the slide arrow may be represented numerically using its components in the x-direction and y-direction. For instance, a slide arrow BB’ may be represented as the composition of two translations: A translation of 8 units in the x-direction followed by a translation of 4 units in the y-direction (i.e., over 8 units and up 4 units). In other words, the slide arrow BB’ represents a change in the x-direction (i.e., Δx) of 8 units and a change in the y-direction (i.e., Δy) of 4 units.

A similar description could be applied to slide arrows CC’ and AA’, or to any segment joining a point (P, Q) in the interior of the triangle to its corresponding image (P’, Q’) under the translation. In every such case, (P’, Q’) = (P + Δx, Q + Δy). As a consequence, segments joining all pairs of corresponding points have the same slope (Why?). These and other relationships may be explored using the Geometers Sketchpad model 5_Fig5_24.gsp.

Figure 5.24: Translations in Coordinate Geometry

Example 5.9

A translation is characterized by the slide arrow (Δx, Δy) = (5, ^-2). Find the images of the following points under this translation.

a. (0, 0)

b. (4, 9)

c. (^-6, ^-3)

Solution 5.9

a. (0, 0) + (5, ^-2) = (5, ^-2)

b. (4, 9) + (5, ^-2) = (9, 7)

c. (^-6, ^-3) + (5, ^-2) = (^-1, ^-5)

Directed Activity Reflection. Coordinate geometry may also be used to reinforce and deepen concepts related to the transformation of reflection. In Figure 5.25 the presence of a coordinate grid makes is clear that the vertices and their corresponding images are the same distance from the mirror (i.e., the reflecting line). Table 5.2 summarizes this relationship for the vertices and their images under the reflection.

Point	Object Coordinates	Distance to Mirror	Image Coordinates
A	(2, 2)	2	(6, 2)
B	(1, 4)	3	(7, 4)
C	(3, 5)	1	(5, 5)

Table 5.2: Reflection in Coordinate Geometry

Note that, in this case, the x-coordinate of the image point differs from that of its corresponding object point by an amount equivalent to twice the distance of the object point from the mirror. Why? Using the Geometers Sketchpad model 5_Fig5_25.gsp, examine the relationship between the slopes of the segments connecting corresponding points and the slope of the mirror.

Figure 5.25: Reflections in Coordinate Geometry