What is the center of a triangle? This question might be more complicated than you first think it is! In this project, the students can explore this question using interactive geometry software. It is helpful if the students work in small, collaborative groups exploring various ideas they have for what might be the answer to this question.
The exploration can involve different "real-life" applications of the center of a triangle. Here is one of the problems that could be explored: "You are a city planner. The three towns of Kaneohe, Kailua and Maunawili have pooled their funds and want to build a recreation center. Where would you put the rec center so as to be fair to all 3 towns?"
The Geometer's Sketchpad software allows students to explore geometric figures and their relationships. In this activity, the students draw the triangle formed by the three towns. Then they pick a point somewhere inside the triangle to represent "Is this point equally distant from each of the three towns?"
With some experimentation, moving the location of the proposed rec center on the computer, the students find a point that does seem to be equally distant from all three towns. This point, in mathematical terms, the students know to be the Circumcenter, the point where the perpendicular bisectors of the triangle meet.
When I ask the question "Does this seem a fair and reasonable location?", the students agree. But then I ask them another question - "What if the towns were located differently; for example - what if the triangle joining the three towns happened to be a right triangle, or an obtuse triangle? Would this be a reasonable location for the rec center? Is it equally distant from all three towns?" An interesting discussion of "equity" and "fairness" results from these questions.
This project allows students to explore the relationships in mathematics, to look at interesting applications, and to consider all possibilities!
The center of a triangle is a very interesting topic in Geometry. Actually, there is not one but a number of different types of centers; the ones most commonly taught in school are the orthocenter, circumcenter, incenter and centroid. Each of these four centers have interesting properties as described below.
The orthocenter of a triangle is the point at which the altitudes meet. The orthocenter lies in different places in different triangles. The orthocenter is interesting because in an acute triangle, the orthocenter lies inside the triangle, for a right triangle, the orthocenter lies on the right angle vertex of the triangle, and for an obtuse triangle, the orthocenter lies outside the triangle.
The Circumcenter of a triangle is the point at which the perpendicular bisectors of the sides of the triangle meet. The circumcenter is interesting because it is the center of a circumscribed circle.
The Incenter of a triangle is the point where the bisectors of the 3 angles of the triangle meet. What is interesting about the incenter is that the center of an inscribed triangle.
The Centroid of a triangle is the point at which the medians of the triangle meet. Since a median is a segment, and always in the interior of the triangle, the centroid is always inside the triangle as well.
In this project, we will explore the most interesting property of the centroid: it is also the center of balance, or center of gravity of the triangle! There are 2 methods teachers can use:
Method 1: Give each student a randomly shaped scrap of lightweight cardboard ("poster board" or "railroad board") and ask them to use a straightedge to draw a triangle, and then a compass and straightedge to construct all 3 medians. They should then cut out the triangle with scissors.
Method 2: Give each student a pre-cut triangle (all the triangles should be of different sizes and shapes), and ask them to draw all 3 medians. They can find the midpoint easily by folding a vertex to another vertex and mark the midpoint by "pinching" the index paper.
When they have completed their triangles, students will find that if they push the sharp point of a pencil slightly into the centroid, they can balance their triangle on this point.
An interesting project, if time allows, is to create a "mobile" of triangles, each hanging from its centroid. Give each student a 8 to 10 inch piece of string or fishing line, and a metal "brad". You may use a tack, but this will require you to bend the tack point over with a pair of pliers to form a small loop as shown below:
The metal brads might be easier to bend. Each student can hang their triangle by pushing the metal tabs of the brad, or the bent point of the tack, through the centroid of their triangle. If the tabs of the brad or point of the tack are then bent to make a small loop, they can hang their triangle from fishing line or string, and see that it balances on that point.
The class can use wooden rods, metal rods, or sticks to make a mobile combining all of the individual students' triangles.