Location, Location, Location

• Lesson
9-12
1

Students use a dynamic geometry applet to investigate the relationship among the distances from a point inside a regular polygon to each side.

This lesson begins with a personalized version of the somewhat familiar "Surfer and Spotter" problem (Bennett, 2002, p. 82) and extends to regular polygons with an even number of sides.

Group students in pairs. Display the Triangular Island overhead on the projector.

Then, tell the following story:

Suppose that you and a friend have become stranded on a desert island, which is in the shape of a regular triangle. Gilligan is a lazy, incessant bore, so you’d like to build a hut far away from him. In addition, you need to place your hut on the island so that the sum of the distances to each of the three shores will be as small as possible.

Students may wonder why each person would want the sum of the distances to be as small as possible. You can explain that minimizing the sum of the distances makes getting to each side of the island as easy as possible. For realism, state that the island is covered with lots of vegetation; if the sum of the distances is minimized, you will have to clear less vegetation.

In addition, you might also want to tell students that Gilligan is very selfish. Without discussion, he plans to place his hut at the exact center of the island, which means that students will have to place their huts elsewhere. (This is not completely necessary if students work in pairs, as suggested. But if you would prefer that students work individually, this will ensure that not all students place their huts at the center. Later in the lesson, students can compare their locations to Gilligan's location at the center to find out that the sum of the distances is the same.)

Divide the class into pairs, and give each pair a copy of the Triangular Island activity sheet. Tell students they each are to decide where to locate a hut and find the distances to each side. (Eventually, you will want to give each student his or her own copy of the activity sheet. For the moment, however, require students to work together on the same sheet. This will ensure that they choose different locations for their huts, which will make it more likely that they notice that the location does not affect the sum of the three distances.) Note that students may need guidance about how to measure distance to each side. That is, inform them that they should measure the perpendicular distance.

 Triangular Island Activity Sheet

Select several student papers that show different locations for the two huts, and have these students share their ideas with the entire class. Ask, "Which locations seem to be best for the huts? Why?" [Students should notice that the sum of the distances for every location will be the same.]

Follow up by asking, "Is this sum related to any particular characteristic of the triangle?" [This question is difficult and students may just randomly suggest various characteristics—side length, perimeter, area, or height.] Record the possibilities for future reference. This part of the lesson is to initiate thought about the problem and possible characteristics of the triangle. This is not a time to press for exact answers; that will come later. For now, have students put aside their paper-and-pencil work, but they might return to it when they reconsider the list of possibilities.

Hand out the Beyond Triangular Island activity sheet. With the Triangle Island applet, students can explore the distance to the sides for equilateral triangles and squares. With the Hexagon Island applet, students can explore hexagons. To explore octagons, students will have to draw the figures themselves and find the measurements using a ruler or other measuring device; or, geometry software such as Geometer's SketchPad® can be used.

 Beyond Triangular Island Activity Sheet

You may need to provide directions to students about how to locate the applets and how to use them. Project the URL of the applets on the overhead projector, write them on the chalkboard, or tell students to do the following:

• Go to the home page of the Illuminations web site (http://illuminations.nctm.org).
• Go to the Activities section.
• Click on "View All Activities," or press the Search button.
• From the list of activities, scroll to the link for Triangle Island, Hexagon Island, or Octagon Island.

Once the applets have been found, students can click on the plus sign in front of Instructions to learn how to use the applet. The applets may be used in a whole-group setting or by pairs of small groups of students in a lab setting, depending on the availability of technology.

 Triangle Island Activity

 Hexagon Island Activity

 Octagon Island Activity

As students gather the first set of measures for a triangle, verify that they are able to correctly manipulate the applet. The applet allows students to explore triangles of different sizes, something that would be difficult if students continued to use only paper triangles. (Note that students were never told how big the island was, so exploring multiple sizes is important.) To complete Question 1 on the activity sheet, allow students to gather data for triangles of other sizes.

In whole-group discussion, students should formulate conjectures based on the measures they accumulated. Ask, "What conjecture can you make about the sum of the distances to each side of the triangle for any location of the hut?" [For different locations of the interior point, the sum of the distances to each side of the triangle is constant.] Then ask, "Is this sum related to any particular characteristic of the triangle? Explain your reasoning." [The sum appears to be equal to the height of the triangle. Students may reason through this observation by considering what happens when the hut is at a vertex of the triangle. Dragging the interior point to each of the vertices of the triangle, students may note that the distances to two of the sides become zero while the distance to the third side becomes the height of the triangle.]

Return to the paper triangles. Ask, "How do your results from the applet match your earlier results with the paper triangles?" [Students may note that the two sets of results are similar, so long as their measurements for the paper triangle were accurate.]

Point out to students that the island could have other shapes. It could be square, or it could have more that four sides. Ask students to consider a few different cases. In order to have a class set of data from which students can generalize a relationship for all regular polygons with an even number of sides, assign one of the remaining three polygons (square, hexagon, octagon) to each pair of students. Each pair starts with its assigned polygon and then moves through the other two polygons, time permitting. For each polygon, the pair must complete the table and respond to the questions on the activity sheet. [For the square, the sum of the distances appears to equal twice the height of the square; for the hexagon, the sum appears to equal three times the height of the hexagon; and for the octagon, the sum appears to equal four times the height of the octagon.]

After students have had an opportunity to consider at least one of the square, hexagon, or octagon, and depending on time, the students return to a whole-group setting. A student from a pair originally assigned to each polygon shares the pair’s results. Other students are responsible for questioning the pair about their reasoning.

When the results for all of the polygons are public, pose the following question: "How might you generalize your results to islands of other shapes?" If this question is too general, students’ thinking may be supported by asking them to think about a concrete case involving a regular polygon with a large even number of sides. For example, what would be the relationship between the sum of the distances to the sides of a 50‑gon and the height of the 50‑gon? [The sum will be equal to 25 times the height of the 50‑gon.] In general, how can this observation be stated, that is, in terms of an n‑gon? [The sum will be equal to n/2 times the height of the n‑gon. Students should see that this generalization applies only to n‑gons with an even number of sides. Different relationships exist for polygons with an odd number of sides.]

Assessments

Students create and video record their own explanations. These explanations could be available for a parent night, math day, students’ (electronic) portfolios, or some other event in which visitors (as well as the teacher) will get to hear and see them.

Extensions

1. For the case of the regular triangle, the sum of the distances from any interior point to the three sides is equal to the height of the triangle. Prove this result.
[Subdivide the triangle into three smaller triangles, as shown below.

Because the triangle is equilateral, AB = BC = CA. The area of the yellow triangle is ½(AB)(HR); the area of the green triangle is ½(BC)(HQ); and, the area of the red triangle is ½(CA)(HP). Consequently, the area of the triangle is ½(AB)(HR) + ½(BC)(HQ) + ½(CA)(HP). But because the sides are equal, this can be rewritten as ½(AB)(HR + HQ + HP).

In addition, the area of the triangle is equal to ½(height)(base) = ½(h)(AB).

Setting these two results equal gives:

½(AB)(HR + HQ + HP) = ½(h)(AB)

Dividing both sides by ½(AB) gives HR + HQ + HP = h, so the sum of the distances to each side is equal to the height of the triangle.

Note that the interior point in the image above appears to be located at the exact center of the triangle. This is not necessary for the proof; in fact, note that the explanation does not rely on the location of the interior point. Consequently, the proof works in the general case.]

2. What does this observation tell us about the relationship between the height of an equilateral triangle and the diameter of its circumscribed circle?
[The distance from the circumcenter to the vertices is the radius, r, of the circumcircle, as shown in the figure. The circumcenter is the same distance, s, from each side. If h is the height of the triangle, then 3s = h. Further, r = 2s.

Putting these pieces together gives h/3 = r/2, or r = 2h/3. Consequently, the diameter of the circle is 2r = 4h/3.]

Questions for Students

1. Why does it make sense that the sum of the distances from an interior point to the sides of a regular polygon with an even number of sides is a multiple of the height of the polygon?

[Since the regular polygons with even number of sides will have pairs of sides parallel, the combined distance for each these pair will equal the height of the polygon. Also, the number of pairs will be one‑the number of sides of the polygon. So, the sum will always be

n/2 times the height of the polygon. The square portion of the Triangle Island

applet can be used to show this for the specific case of a four‑sided polygon, which may help students see that it would be true for other even‑sided polygons.]

Teacher Reflection

• What characteristics of regular polygons did the students discuss? Why might these characteristics be important to the students?
• Did your students suggest a relationship between the sum of the distances to each side of the polygon and a characteristic other than the height? How might they follow up on their suggestion using a dynamic geometry program to simulate the problem?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

Learning Objectives

Students will:

• Conjecture the best location of a point inside a regular polygon such that the sum of the distances to each side is a minimum.
• Articulate the relationship between the minimum sum of the distances to each side of a regular polygon and characteristics of the polygon (i.e., height).
• Generalize the relationship between the minimum sum of the distances to each side of a regular polygon and characteristics of the polygon for any regular n‑gon, where n is even.