Copy of Exploring Triangle Centers in Depth
Centroid, Orthocenter, Incenter, Circumcenter
Observe the relationships of special segments of triangles and their points of concurrency. Includes medians, altitudes, angle bisectors, perpendicular bisectors, inscribed circle, circumscribed circle, and the Euler line.
Instructions: Answer all of the questions. You may either type your answers in a text editor and then print, or you may write your responses in notebook paper.
- Which triangle centers always stay in the interior of Triangle ABC? ~The centroid and Incenter always stay inside the triangle.
- When all the centers are in the interior, is the triangle acute, right, or obtuse? ~The triangle is acute when all centers are inside the triangle.
- Where is the location of the orthocenter when Triangle ABC is formed into a right triangle? ~The orthocenter is on the right angle when the triangle is right.
- Which triangle centers are always collinear? ~Centroid, circumcenter, and orthocenter are always colliner on the euler line.
- View the Side Lengths and try to make an equilateral triangle. What appears to happen to the triangle centers? ~When the triangle is equilateral all the centers form align on the euler line.
- View the Medians, Altitudes, Angle Bisectors, and Perpendicular Bisectors. Which of these lines always pass through the Midpoints? ~Perpendicular bisectors and angle bisectors both pass through the midpoint of each side.
- View only the Altitudes and Perpendicular Bisectors. How are these lines related? ~Both intercept the sides of the triangle at 90 degree angles.
- View the Circumcircle (Circle ABC) and drag point A across segment BC. What happens to the Circumcircle? ~The circle becomes larger then smaller then smaller again. It becomes larger as it passes segment BC
- Compare and contrast the Circumcircle and the Inscribed Circle ~The inscribed circle is always inside the triangle and the circumcircle is always outside the triangle. Both however always touch all 3 of the sides of the triangle.
- View the Inscribed Circle, Angle Bisectors and the Radii of Inscribed Circle. Look closely at how they meet Triangle ABC. a. Does the Inscribed Circle reach where the Angle Bisectors intersect with Triangle ABC? ~The inscribed circle reaches where the angle bisector intercepts the line but not the angle itself. b. At what angle do the radii meet Triangle ABC? ~The radii meets triangle ABC at a 90 degree angle.
- Think about cutting out the triangle and balancing it flat on your finger. Which triangle center evenly distributes the area (or weight) of Triangle ABC? In other words, where is the center of gravity? Explain your reasoning. ~The center of gravity is the centeroid triangle because it uses all of the midpoints.
- Now think of points A, B, and C as cities, and we want to build a hospital in the middle that has new Life Flight helicopters. We want to find the center location that is equidistant from all three cities. Which triangle center represents the point that is the same distance from each vertex of Triangle ABC? How can you be sure? ~I know that the incernter triangle is in the middle because it is where all the angle bisectors meet.
- Next, think of segments AB, BC, and CA as major highways, and we want to build a fire station in the middle with three roads leading to the highways. We need to build the quickest access routes to the highways in order to reduce the amount of time it takes to for the fire trucks to get on the highways. Which triangle center represents the point that is the least distance from each side of Triangle ABC? How can you be sure? ~The point that is the least distance from each of the sides is the centroid triangle because it is where all the medians intercept each other.
- Finally, wondering what is the purpose of the Orthocenter? (Hint: you should have used the other three triangle centers for the 3 previous questions.) Show the Inner Triangles and make Triangle ABC acute. Which inner triangle has the smallest perimeter? Walking the edge of this inner triangle is the shortest path between the segments AB, BC, and CA. Can you think of a situation in real life when you might want to find the shortest path between the sides of a triangle? ~The purpose of the orthocenter is to help you find the altitudes. The orthocenter is the inner triangle with the smallest perimeter. You might want to find the shortest path between streets when you are running late and need to get there faster.
- Which of the inner triangles is similar to Triangle ABC? ~The centeroid triangle is most like triangle ABC.
- Is the Centroid of Triangle ABC also the centroid of the inner centroid triangle ~Yes.
- What would happen if you continued to draw centroid triangles inside centroid triangles? Would they always be similar?