Explore Triangle Centers
Explore Triangle Centers
Centroid, Orthocenter, Incenter, Circumcenter
Observe the relationships of special segments of triangles and their points of concurrency.
Instructions: Answer all of the questions. Submit your answers in Canvas via a google doc.
Part 1:
1) Create an acute triangle by dragging one point on a given triangle.
2) Click "Median". Where are medians passing through on a given triangle?
3) What is the name of the center point?
4) Change the acute triangle to the right triangle. Where is the centroid?
5) Change the right triangle to an obtuse triangle. Where is the centroid?
6) Refresh the GeoGebra board.
Part 2:
1) Create an acute triangle by dragging one point on a given triangle.
2) Click "Altitude". Where are altitudes passing through on a given triangle?
3) What is the name of the center point?
4) Change the acute triangle to the right triangle. Where is the orthocenter?
5) Change the right triangle to an obtuse triangle. Where is the orthocenter?
6) Refresh the GeoGebra board.
Part 3:
1) Create an acute triangle by dragging one point on a given triangle.
2) Click "Angle Bisector". Where are angle bisectors passing through on a given triangle?
3) What is the name of the center point?
4) Change the acute triangle to the right triangle. Where is the incenter?
5) Change the right triangle to an obtuse triangle. Where is the incenter?
6) Click "Inscribed Circle". Where is the location of a circle? Does it change the location as you are changing a triangle to acute, right, or obtuse?
7) Refresh the GeoGebra board.
Part 4:
1) Create an acute triangle by dragging one point on a given triangle.
2) Click "Perpendicular Bisector". Where are perpendicular bisectors passing through on a given triangle?
3) What is the name of the center point?
4) Change the acute triangle to the right triangle. Where is the circumcenter?
5) Change the right triangle to an obtuse triangle. Where is the circumcenter?
6) Click "Circumcenter Circle". Where is the location of a circle? Does it change the location as you are changing a triangle to acute, right, or obtuse?
7) Refresh the GeoGebra board.
Q/A
1) When all the centers are in the interior, is the triangle acute, right, or obtuse?
2) Where is the location of the orthocenter when Triangle ABC is formed into a right triangle?
3) Where is the location of the circumcenter when Triangle ABC is formed into a right triangle?
5) Click "Euler Line". Which triangle centers are always collinear?
6) Click "Side Lengths" and try to make an equilateral triangle. What appears to happen to the triangle centers?
7) Click "Ratio found in Euler Line". The ratio of the lengths between those centers is constant. What is that ratio? (Write a proportion: an equation of the ratio of the segments equal to the ratio using numbers).
8) Click the Medians, Altitudes, Angle Bisectors, and Perpendicular Bisectors. Which of these lines always pass through the Midpoints?
9) Click only the Altitudes and Perpendicular Bisectors. How are these lines related?
10) The centroid is special in that for each median, the ratio is the same when comparing the length of the segment from a vertex to the centroid to the length of the median from that vertex. What is that ratio?
Challenging Questions
1) 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.
2) 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?
3) 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 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?