Teaching the Perpendicular Bisector: A Kinesthetic Approach

Teaching the Perpendicular Bisector: A Kinesthetic ApproachAuthor(s): Ayana TouvalReviewed work(s):Source: The Mathematics Teacher, Vol. 105, No. 4 (November 2011), pp. 269-273Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/10.5951/mathteacher.105.4.0269 .

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Vol. 105, No. 4 • November 2011 | MATHEMATICS TEACHER 269

This series of sequential activities has proved particularly useful with students who are already familiar with such concepts as the midpoint of a segment, perpendicular lines, and the perpendicular bisector. Indeed, I like to introduce these activities only after students have had some initial practice in drawing various examples of perpendicular bisectors. These activities are appropriate for both middle school and high school geometry classes.

Through movement—a welcome change of pace—students explore the properties of the perpendicular bisector.

Ayana Touval

Kinesthetic intelligence is one of the seven kinds of intelligence identified by Gardner’s multiple intelligence theory (1983). The kinesthetic approach to teaching has numerous

pedagogical advantages and can be adapted to the teaching of mathematics. This article describes a series of kinesthetic activities designed to explore the properties of the perpendicular bisector.

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PerpendicularPerpendicularPerpendicularPerpendicularTeaching the

Copyright © 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.



270 MATHEMATICS TEACHER | Vol. 105, No. 4 • November 2011

firmed that he or she was apparently equidistant from points A and B.

To check this premise, I asked one student who stood on the perpendicular bisector line to mark the point on which he was standing with a piece of masking tape and to count his whole or par-tial steps to the student standing on point A. The student then repeated the exercise by counting his steps to the student standing on point B. The exercise confirmed students’ observation that the distances are equal.

After deliberating, we phrased this finding as follows and recorded it on the board: “All the points on the perpendicular bisector of a segment are equi-distant from the ends of the segment” (see fig. 1).

At this point, I asked students to try to state the converse of the theorem. After some deliberations, a student came up with the following question: “Is a point that is equidistant from the endpoints of a segment AB located on its perpendicular bisector?”We decided to seek the answer through another activity.

ACTIVITY 3: EXPLORING THE COMMON PROPERTY OF POINTS THAT ARE EQUIDISTANT FROM THE ENDPOINTS OF A SEGMENT One student volunteered to stand on point A and one to stand on point B. Then eight pairs of ribbons were divided equally between them, with each stu-dent holding one end of every pair.

I then asked a third student to pick from the students at points A and B the loose ends of a pair of ribbons having the same color and move around until both ribbons were pulled taut. Seven other students followed suit, each picking the loose end

ACTIVITY 1: FINDING THE PERPENDICULAR BISECTOR BY WALKINGThis task required, first, that the students create a segment. To do so, a student placed a piece of masking tape on the classroom floor (see sidebar “Required Materials”) and marked one end of it with the letter A and the other end with the letter B to indicate endpoints. The tape should be approxi-mately 5 feet long to allow enough length for stu-dents to count off their steps.

To find the midpoint, another student walked from point A to point B, counting her steps and marking the midpoint with the letter C. Stand-ing on point C, the student placed her feet in the direction of point B. She then made a 90° turn and walked on the ray perpendicular to the segment. As she did so, students commented that she could have turned in the opposite direction. The walk demonstrated to students that the perpendicular bisector is a line that can lie on either side of the segment.

Several students repeated the process by identi-fying the midpoint, making a 90° turn, and walking on the perpendicular line. The exercise was com-pleted by a student placing a long piece of mask-ing tape on the floor to mark the perpendicular bisector.

ACTIVITY 2: ALL POINTS ON THE PERPENDICULAR BISECTOR ARE EQUIDISTANT FROM THE ENDS OF THE SEGMENT After marking with tape the perpendicular bisec-tor of the segment with endpoints A and B, I asked students if there was anything special about this line. To explore this question, I asked one student to stand on point A and another to stand on point B. A third student then placed himself on the per-pendicular bisector line and immediately observed that his distance to the student on point A and to the student on point B was equal.

Other students then verified this observation by placing themselves on different points along the perpendicular bisector line. Each one then con-

Fig. 1 Students fi rst eyeballed and then measured to

conclude that segments AD and BD were equal in length.

Required Materials Activities 1–5: Masking tape

Activity 3: At least 8 pairs of narrow, colored ribbons, each pair a

different length and color; the length of the shortest pair should be

half the distance of segment AB

Activities 4–5: A long string about 12 feet long with ends tied

together to form a loop; an additional 5-foot piece of string




of another pair of matching ribbons and walking until finding the spot where the ribbons they were holding were pulled taut.

Invariably, all eight students found themselves standing on the perpendicular bisector. They were delighted by this discovery (see fig. 2).

We then wrote our finding on the board, phras-ing it this way: “Points that are equidistant from the endpoints of a segment AB lie on its perpen-dicular bisector.”

Activities 2 and 3 opened a lively discussion about a theorem and its converse. All the students observed that in activity 2 the perpendicular bi-sector was given, whereas in activity 3 the points were given. The colored ribbons made these activities all the more memorable and later helped students greatly when they examined lines in the triangle, especially the perpendicular bisectors in a triangle.

ACTIVITY 4: INVESTIGATING WHETHER THE THREE PERPENDICULAR BISECTORS OF THE SIDES OF A TRIANGLE ARE CONCURRENT The next exercise took us further in our explo-ration of the perpendicular bisector and helped answer the following question: “Do the perpendicu-lar bisectors of the three sides of a triangle meet at a single point?” Initially, some students seemed skeptical about this possibility. We decided to form a triangle and explore its perpendicular bisectors kinesthetically.

Three students volunteered to represent the vertices of triangle ABC. Holding the long loop of string tightly among the three of them, they placed it on the floor. The triangle they created was

acute. Each vertex was taped to the floor to help keep the triangle’s shape.

Three other students were asked to walk along the perpendicular bisectors of the three sides. To do so, each student identified the midpoint of one of the three sides, made a 90° turn, and walked in the direction of the inside of the triangle. Very shortly, the three students met. It appeared that we could conclude that the three perpendicular bisectors of a triangle intersect.

Was this rule valid for all triangles? To answer this question, several students immediately sug-gested that we change the acute triangle to an obtuse triangle. This activity encouraged much dis-cussion among the students.

After creating the obtuse triangle with the long string, the students who volunteered to walk along the perpendicular bisectors wondered in which direction to walk now. After making some initial steps in various directions, they found themselves meeting on a single point outside the triangle. They marked this point on the floor with a piece of mask-ing tape with the letter D on it.

As we all looked at point D, I asked students whether they could identify anything special about it. “It is on all three of the perpendicular bisectors,” said one student. “So?” asked another. And then a third student commented: “That means that this point must also be equidistant from all the verti-ces!” This was a major observation.

To verify this observation, a student stood on point D. She asked another student holding a piece of string to join her. The student standing on point D took one end of the string and pulled it taut. She then asked the second student to walk the distance from point D to vertex A. He did, and as he reached vertex A, he pulled the string taut and began walk-ing around the triangle. After a few moments, the

Fig. 2 When the colored ribbons were taut, the student

was standing on the perpendicular bisector.

Fig. 3 Point D, marked by tape on the classroom fl oor,

represented the intersection of the perpendicular bisectors

of the sides of the triangle.



272 MATHEMATICS TEACHER | Vol. 105, No. 4 • November 2011

student crossed all three vertices. As he continued walking, it became evident to everyone that he was walking in a circle.

We gave this circle its name—the circumscribing circle—and wrote on the board the following result: “The perpendicular bisectors of the sides of a tri-angle intersect at the center of the circumscribing circle of the triangle” (see fig. 3).

ACTIVITY 5: EXPLORING SOME PROPERTIES OF PERPENDICULAR BISECTORS IN A RIGHT TRIANGLEWould the lesson learned in activity 4 hold also for a right triangle? To work through this problem, three students created a right triangle with the loop of string. Taping down the vertices A, B, and C, they placed the string triangle on the floor. Finding the intersection of the perpendicular bisectors in this tri-angle proved particularly gratifying because students realized that the intersection lay on the hypotenuse. Students wanted to create the circumscribing circle.

Two students volunteered to form the circle. One student stood on the midpoint of the hypot-enuse while the other, holding the end of another piece of string that spanned the distance from the midpoint of the hypotenuse to one vertex, walked around the triangle to create a circle.

ACTIVITY 6: RECREATING A CIRCLE FROM A GIVEN ARC USING KNOWN PROPERTIES OF THE PERPENDICULAR BISECTOR Teaching geometry offers many opportunities to address real-life applications of theoretical mate-rial. The next lesson provided a nice example of an application of the preceding work.

This activity required that students be divided into two groups and that one group leave the class-room. The remaining students then used a piece of string to begin forming a circle on the floor. To accomplish this task, they arbitrarily marked a point on the floor with a piece of tape. One student then stepped on this spot, holding one end of the

string. Another student picked up the other end, pulled it taut, and began walking, taking care to keep the string taut. It was clear to everyone that she was walking in a circle. As she walked, another student followed her movement and stuck pieces of tape on the floor to mark points on her path. Gradually, students saw that the pieces of tape on the floor created a visible arc (see fig. 4).

We then removed the tape marking the center of the circle and invited the other students back into the classroom. As they entered the room, they saw a number of pieces of tape on the floor that formed the shape of an arc of a circle. Their task was two-fold: to find the center of the lost circle and to mea-sure the length of its radius.

Almost immediately, one student suggested the following process: He would choose three random points on the arc and form a triangle (see fig. 5). Then, putting into practice the lessons of the previ-ous activities, he would identify the midpoint of the three segments and walk along their perpendicular bisectors to find the point of intersection. Three students volunteered to join him in finding the three perpendicular bisectors and walking along them to reach the intersection.

As they began walking, another student sug-gested that it might not be necessary to find all three bisectors—that two, in fact, might be enough. The three students checked whether this was true and concluded that the intersection of two perpen-dicular bisectors must also be the intersection of all three. As one student explained, the point at which two perpendicular bisectors meet is equidistant from all the vertices. The intersection was clearly the center of the lost circle on the floor.

Having confirmed this principle, the students found that the length of the radius was equal to the distance between the center of the circle and one of the vertices of the triangle.

SUMMARYThe activities presented here have proved to be an instrumental pedagogical tool in my geometry classes. Needless to say, they were not conducted in a vacuum; rather, they were incorporated into class

Fig. 5 Inscribing a triangle in the arc was a fi rst step to

rediscovering the circle’s center.

Fig. 4 Students used tape to mark points on the circle’s arc.




lessons in which students attempted to prove theorems and connect geometrical concepts.

The perpendicular bisector lesson was well remembered throughout the year. Students who had had difficulties distinguishing between a theorem and its converse found activities 2 and 3 especially helpful. These activities also paved the way for students to understand the properties of the angle bisector and ultimately the idea of locus. All in all, these activities provided a welcome change of pace and, for some students, a unique opportunity to par-ticipate and be actively involved in class discussion.

ACKNOWLEDGMENTSI wish to thank Frances Thompson, who helped me revise this manuscript, and Galit Ben Zion, with whom I have collaborated in creating many kines-thetic activities.

BIBLIOGRAPHY Gardner, Howard. 1983. Multiple Intelligence: The

Theory in Practice. New York: Harper Collins. Touval, Ayana. 2009. “Walking a Radian.” Mathemat-

ics Teacher 102 (9): 692–96.Touval, Ayana, and Galeet Westreich. 2003. “Teach-

ing Sums of Angle Measures: A Kinesthetic Approach.” Mathematics Teacher 96 (4): 230–37.

AYANA TOUVAL, [email protected], teaches at Montgomery College in Rockville, Maryland. Throughout her teaching

career in high school and community college, she has explored ways to share her love of mathematics with her students.

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Teaching the Perpendicular Bisector: A Kinesthetic Approach