Constructing Points of Concurrency
- Leigh Anne
Investigation 1: Angle Bisectors in a Triangle
Step 1 Construct the angle bisector of angle A. Step 2 Construct the bisector of angle B. Step 3 Construct point D where these two rays intersect. Investigate 1. Construct the bisector of angle C. What do you notice about this third angle bisector? Drag the vertices of the triangle to see if this is always true. 2. If you were blindfolded and drew three lines on a piece of paper, do you think they would intersect in a single point? When lines that lie in the same plane do intersect in a single point, they’re said to be concurrent. Complete the Angle Bisector Concurrency Conjecture (C-9) in your conjecture packet. 3. The point of concurrency of the angle bisectors in a triangle is called the incenter. Measure the distance from point D to AB. How did you do this? Also measure the distances from point D to the other two sides. Drag parts of the triangle; complete the Incenter Conjecture (C-13) in your conjecture packet.
Investigation 2: Perpendicular Bisectors in a Triangle
Step 1 Construct points D and E, the midpoints of segment AB and segment AC. Step 2 Construct the perpendicular bisector of segment AB. Step 3 Construct the perpendicular bisector of segment AC. Step 4 Construct point F where these two perpendicular bisectors intersect, and then construct the perpendicular bisector of segment BC. Investigate 1. Complete the Perpendicular Bisector Concurrency Conjecture (C-10) in your conjecture packet. 2. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter. Measure distances from the circumcenter to each of the three vertices of the triangle. Drag parts of the triangle; complete the Circumcenter Conjecture (C-12) in your conjecture packet.
Investigation 3: Altitudes in a Triangle
Step 1 Construct the perpendicular line from point B to side AC. This line contains an altitude of triangle ABC. Step 2 Construct a line through point A that contains an altitude. Step 3 Construct point D, the point of intersection of these two lines, and then construct the line containing the third altitude in triangle ABC, through point C Investigate 1. Complete the Altitude Concurrency Conjecture (C-11) in your conjecture packet.
Investigation 4: Conjecture About Medians
Step 1 Construct the midpoint D of AC and median BD. Step 2 Construct the midpoint E of AB and median CE. Step 3 Construct point F where these medians intersect, and then construct the third median, from point A to side BC. Investigate 1. Complete the Median Concurrency Conjecture (C-14) in your conjecture packet.
Investigation 5: The Centroid Conjecture
Step 1 Change the label of point F to Ce, for centroid. The centroid is the point of concurrency of the medians. Step 2 Measure the distance from B to Ce and the distance from Ce to D. Step 3 Drag vertices of triangle ABC. Look for a relationship between BCe and CeD. Step 4 Make a table of at least FIVE distance measurements on a separate piece of paper. Step 5 Move a vertex of the triangle to enter the distance measurements into your table. Investigate 1. Keep changing the triangle and adding entries to your table until you find a relationship between the distances BCe and CeD that holds for any triangle. Complete the Centroid Conjecture in your conjecture packet.