The following problem can be used as a brief introduction to this lesson:
Suppose a soccer player is running down the sideline on a breakaway, and she wants to take a shot. At what point should she shoot to give herself the best chance of scoring the goal?
The mathematical problem represented by this real-life situation is actually one of determining the point on a line that yields the maximum angle between the point and a perpendicular segment, as shown below.
As it turns out, the optimal angle occurs when the player occupies the location indicated by the point of tangency of a circle that passes through the two goal posts and is tangent to the breakaway line. Although it is possible to construct a circle that is tangent to the breakaway line and passes through the two goal posts, it is much easier to determine the optimal point with a specific application of the Power of Points theorem, which is the focus of this lesson. (The solution to the soccer problem will be covered later in the lesson.)
The soccer problem can be investigated with the Soccer Problem applet. Reducing the problem to its mathematical components, this applet allows students to identify the point at which the maximum angle occurs. You may wish to have students explore this applet before proceeding with the lesson.
The Power of Points Theorem has three cases: the intersection of two chords, the intersection of two secants, and the intersection of a tangent and a secant. These three cases are typically presented as three separate theorems in most secondary geometry textbooks. This lesson presents these cases within one theorem so that students may view the cases as one unified concept.
Show students the Chord Problem overhead. Ask, "How would you determine the value of y?" [Students might suggest measuring the segment, to which you might respond with a question about how the student would handle the accuracy of the measurement. (It is best to not give copies of the overhead to students to encourage reasoning rather than measurement.) Students may also suggest using similar triangles. Follow this line of reasoning and then question students about whether this method will work in all cases. Students might suggest various combinations of operations with the given numbers. These approaches will likely lead to contradictory answers. Record these conflicting answers without judgment. Without affirming or rejecting any suggestions, proceed to the next part of the lesson as a way for students to determine which, if any, of their suggestions are valid.]
At this point, the Power of Points applet may be used in a whole-group setting or by pairs or small groups of students in a lab setting, depending on the availability of technology. While it is best to allow students to explore the applet on their own, it is also possible to use it as a demonstration tool, displaying the applet with a projection device.
Give students the following directions: "Drag point P to various locations within the interior of the circle. Observe the product of the lengths of the two segments for each chord. Write conjecture(s) based on your observations. Drag other points, e.g., points A, C, or G. Does the conjecture still hold?" [Students should note that the products are equal; in addition, the products become smaller as the point is nearer the circle. Dragging other points does not change the equality of the products.]
Ask students to consider the approaches that were suggested for the Chord Problem in light of this conjecture. [Students should be able to discount or support some of the suggested approaches. They will return to the Chord Problem later in the lesson.]
Ask students to now consider what happens if point P is moved outside of the circle. Tell students, "Now drag point P into the exterior of the circle and describe what happens." Students likely will see two secants. If they see a combination of a tangent and a secant, there is no need to emphasize that combination as a special case. Ask students, "Do your conjecture(s) for interior points also hold for exterior points? If necessary, modify your conjecture(s) so that your conjecture(s) would be true for both interior points and exterior points. If your conjecture holds when you drag P, does it also hold when you drag other points?" [Conjectures about the products being equal will still be valid.]
By this time, students will have considered points within the circle and outside the circle. Ask them, "What locations of point P have you not yet considered?" Students should note that points lying directly on the circle have not been considered. For points on the circle, students may notice that the common product is 0; the Near Circle Extension relates to this observation.
Direct students to move point P to a location outside the circle until one of the secants is tangent to the circle. It may be a little tricky to actually locate the tangent using the applet. The point is that two points will meet at the exact same spot when the tangent is identified. (Depending on which secant students attempt to modify, either A and B will overlap or C and D will overlap when the point of tangency is located.) Students should then realize that two segments from P to the point of tangency are coincident, which is why the same conjectures will still hold. Ask students, "Do your conjecture(s) still hold? If necessary, modify your conjecture(s). If you believe your conjecture holds for different locations of P, does it hold when you drag other points?" [The products are still equal.] At this time, students may want to move point P until they observe two tangents to the circle; technically, this may be difficult to accomplish with the applet, although it is possible.
Applying the tangent-secant case of the Power of Points theorem to the soccer problem introduced at the beginning of this lesson will result in the desired solution. The optimum shot occurs when the location of the player is the point of tangency for the circle that passes through the goal posts and is tangent to the breakaway line:
To calculate the position of the player for the best shot, P, apply the tangent-secant case: PD2 = AD × BD, and solve for PD.
At this point, students often realize that the products are equal in all cases. The lesson now turns to justifying the conjecture for the general case.
The three diagrams below show how segments can be added to each of the three cases to form triangles. In each diagram below, ΔPAD is similar to ΔPCB. As a result, it can be shown in each case that AP × BP = CP × DP. (Note that in the tangent‑secant case (the right‑most figure below), points C and D are concurrent.)
Explain to students, "The observed product for a particular location of point P is called ‘the power of point P’ with respect to the given circle. We have associated this power of a point with the common product of two pairs of segment lengths. How can you use this fact to solve the original Chord Problem?" [Students should be able to compute a value that would be the solution for y in 5.18 × 2.52 = y × 6.73, which comes from wx = yz.]
There are three numerical problems demonstrating several different powers for P. Display the problems on the Numerical Problems overhead one at a time. Have students work in pairs or individually to solve these problems. After students have solved all three problems, ask, "The power of P in these problems are quite different, yet the values for PD are very similar. How can you explain this?" [Students may notice that the circles are different sizes and have different diameters. The value of PD might be the same if the circles were the same size.]
To conclude the lesson, ask one or more students to describe how the different locations of point P relate to each other and how this relationship makes solving three kinds of numerical problems like solving one type of numerical problem. [All locations of point P involve a circle, two segments, and the point of intersection of these two segments. The length of each of these segments is expressed as either the sum or difference of a pair of segment lengths. The product of the lengths of the segments in one pair is the same as the product of the lengths of the segments in the other pair. Solving any of the three types of problems is a matter of knowing there is a common product and calculating that value.]
