Journey to the Center of a Triangle: Answers!

Are you sure you want to know the answers to the 3 questions asked in our "Journey to the Center of a Triangle"? Do you want to explore it some more before you read on? If not, here are the long-awaited answers to the questions!

Question 1: You are a city planner. The three towns of Kaneohe, Kailua and Maunawili have pooled their funds and want to build a recreation center. The 3 towns are shown in the map below. Where would you put the rec center so as to be fair to all 3 towns?

Answer: You want the rec center to be equally distant from each of the 3 towns, right? The set of points equidistant from 2 given points is the perpendicular bisector of the segment joining the 2 points, so the point where the perpendicular bisectors meet will be equidistant from all 3 towns. This point is the center of a circle circumscribed around the triangle formed by the 3 towns. This point is called the circumcenter of the triangle.

Question 2: You are a sculptor and have just completed a large metal mobile. You want to hang this mobile, made of flat triangular metal plates, in the State Capitol. Each triangular piece will hang so that it will be suspended with the triangular surface parallel to the ground. From what point should each piece hang?

Answer: In order to hang horizontally, parallel to the ground, each triangle must hang from a point called the Centroid. The centroid is the point where the medians of the triangle meet. If we look at one median of a triangle, segment AD below, we can see that this segment divides the original triangle into two triangles which have equal bases (BD and CD) and the two triangles share an altitude (AE). (Recall that the altitude of a triangle is a segment from a vertex perpendicular to the opposite side, and that 2 of the altitudes of an obtuse triangle fall outside the triangle.) This makes the 2 triangles, ABD and ACD equal in area. They are not necessarily congruent triangles.

If we construct all 3 medians of a triangle, we will find 6 triangles which are all equal in area, for reasons related to the discussion above. Therefore, the point where the 6 triangles meet, point G, is the point which is the center of balance or center of Gravity for the original triangle: the point around which the area is equally distributed, if the triangle is of uniform thickness. Therefore this is the point from which to hang the triangular metal plate so that the mobile will balance.

Question 3: You are an architect. You are designing a round office tower. The client wants the tower to be circular. The property on which it will be built is at the corner of two streets and there is another buildng on the property Where would you place the center of the lanai, to make the lanai as large as possible?

Answer: The center of the round house needs to be equidistant from the street and the two properties lines: if we must therefore be equidistant from the 3 sides of the triangle formed by the 3 lines. If we consider 2 lines at a time, the set of points equidistant from two lines is the bisector of the angle between 2 lines. So the set of points equidistant from all 3 lines is the point where all 3 angle bisectors meet. This point is called the Incenter of the triangle. Because it is equidistant from the 3 sides of the triangle, if we use that distance as radius, we can construct a circle tangent to all 3 sides of the triangle. This circle is the outline of our lovely new round house!

(This architectural model was designed and built by Joseph, in my mechanical drawing class)

If you are interested in finding out more about other applications of the Incenter, click on the link at the bottom of this page.

Another interesting question is "Where are the four centers of a triangle. Where are they located in relationship to each other?" For the answers to these questions, click on the link below:

Where are the 4 centers of a triangle?

There are other applications of the "centers" of a triangle, and the centers of other figures as well. To learn about these ineresting uses of geometry, please click on the link below.

Foreward to Applications of the Center of a Triangle

Back to the Introduction to Journey to the Center of a Triangle