Points and lines, and the relationships between them, are essential ingredients of so many problems in, for example, calculus. In particular, obtaining the equation of the perpendicular bisector of a line segment, dropping a perpendicular from a point to a given line, and calculating the distance from a point to a line are three tasks treated in elementary analytic geometry that recur in the applications. Let's explore the tools that Maple has for implementing these basic calculations.

Table 1 illustrates how to find the equation of the perpendicular bisector of the segment determined by the points  and . The approach incorporates the direct mathematical steps of finding the midpoint and slope of the line segment, and constructing the equation of the perpendicular bisector via the point-slope form of the line.

Table 2 implements the calculation with tools from the Student Precalculus package.

Table 3 shows how to obtain the equation of the perpendicular bisector with the tools of the geometry package. There is a bit of overhead in this package, but the commands available are robust and powerful. If the names of the variables associated with the horizontal and vertical axes are not explicitly declared, then a pop-up will request this information when the Equation command is used to extract the equation of the perpendicular bisector.

It is unfortunate that the data structures for points and lines at top level and in the Precalculus package are not compatible with those in the geometry package. Essentially, that means calculations within the geometry package are isolated from many of the top-level functionalities of Maple, and vice versa. The devil is always in the details.

Table 4 illustrates how to use top-level functionality for dropping a perpendicular from the point  to the line .

Figure 2 could also be created interactively using the Plot Builder accessed from the Context Menu, but this would entail dragging and dropping components onto the final graph. It's more difficult to describe these operations than it is to execute them, so we have opted for the command-based construction of the figure.

Table 5 shows how the calculations in Table 4 could be implemented using tools from the Student Precalculus package. It is not necessary to re-install this package if this has once been done in the session, but for the sake of completeness, this step is repeated.

Table 6 shows how the calculations in Table 4 could be implemented in the geometry package. It is not necessary to re-install this package or to again declare the variables associated with the horizontal and vertical axes if this has once been done in the session, but for the sake of completeness, these steps are repeated.

Although the paradigm for naming in the geometry package is not the norm for Maple, the package provides many useful commands for working in analytic (coordinate) geometry. In particular, the draw command is useful because it will draw the package objects, objects which the ordinary plotting commands in Maple are not designed to graph. Note also how the coordinates of a point of intersection of two lines is returned as a list, unlike the return of the top-level solve command.  It is these small details that determines how easy a calculation will be to implement is a particular piece of software.

Although the distance from the point  to the line  is given by the formula

let's calculate the distance of the point  from the line  from first principles. Hence, we need to drop a perpendicular from  to the line, obtain the coordinates of the point of intersection of the given line with this perpendicular, and compute the distance from these two points.

From Table 4, the point of intersection of the perpendicular from  to the given line is . Hence, the distance  is given by

 =

As a check, the distance formula also gives

 =

The Distance command in the Precalculus package gives

 =

and (using Table 6) the distance command in the geometry package gives

 =

which, of course, simplifies to .

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