Name Date Concurrency where they all meet Geometric Constructions: Circumcenter CIRCUMSCRIBED CIRCLE - Point of concurrency called CIRCUMCENTER. This is the intersection of 3 perpendicular bisectors of each side. Circumcenter - The point of concurrency of the perpendicular bisectors of the sides of a given triangle. Create the perpendicular bisector for each side and extend the line. Where they meet is the circumcenter. The segment from each vertex to the circumcenter are radii of the circles. 1
For the triangle, construct all three perpendicular bisectors to show they are concurrent. Circumscribe a circle about the triangle. Use the intersection of the 3 perpendicular bisectors above. This is the center of the circle. Use the compass to create the circle. 2
INSCRIBED CIRCLE Point of concurrency called INCENTER This is the intersection of 3 angle bisectors (from each vertex). Geometric Constructions: Angle Bisector 3
For the triangle, construct all three angle bisectors to show they are concurrent. Inscribe a circle in the triangle using the center of the circle as where angle bisectors intersect. Use the compass to create the circle. 4
BALANCING POINT Point of concurrency called CENTROID Centroid - The point of concurrency of the medians of a triangle. Connect the midpoint of each side to the opposite vertex. Bisect each side as specified below and then connect it to each vertex. Centroid - The point of concurrency of the medians of each side of a triangle when connected to the opposite vertices of a given triangle (this is called the altitude). Create the bisector for each side and connect the midpoint to the opposite vertex. Where they meet is the centroid. This is the balancing point for the triangle. If you cut out the triangle and put your finger on the centroid, it will balance. Geometric Constructions: Median 5
To create the median, construct a perpendicular bisector of the side opposite A and then connect A. For the triangle, construct all three medians to show they are concurrent. 6
Point of concurrency called ORTHOCENTER Orthocenter - The point of concurrency of the altitudes of a triangle. A For the triangle, construct the altitude from vertex A. a. From point A, use your compass from each side to create a semi-circle on the opposite side. b. From the intersections on each side of the line, draw 2 intersecting arcs. c. Connect this to vertex A and you have an altitude. 7
For the triangle, construct all three altitudes to show they are concurrent. 8
Cell Phone Tower Task A cell service operator plans to build an additional tower so that more of the southern part of Georgia has stronger service. People have complained that they are losing service, so the operator wants to remedy the situation before they lose customers. The service provider looked at the map of Georgia below and decided that the three cities: Albany, Valdosta, or Waycross, were good candidates for the tower. However, some of the planners argued that the cell tower would provide a more powerful signal within the entire area if it were placed somewhere between those three cities. Help the service operator decide on the best location for the cell tower. 1. Just by looking at the map, choose the location that you think will be best for building the tower. Explain your thinking. 2. Now you are going to use some mathematical concepts to help you chose a location for the tower. Using the 4 triangles attached that approximately represent the triangle formed by Albany, Valdosta and Waycross, find the centroid, incenter, circumcenter, and the orthocenter Trace the triangles, mark the point of concurrency, and add it to your map and label it (C, I, CC, O). 3. Choose a location for the tower based on the work you did for question #2. Explain why you choose this point. 4. Compare the point you chose in question #3, based on mathematics, to the point you chose in question #1, based on observation? 9
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Now, you can apply your new found knowledge to an important situation: Because of the possible closing of Grady Hospital, several groups have decided to fund an acute trauma hospital located where it would give the most people in this region of Georgia the most access. Using the cities identified above and the map from the earlier question to decide where the new trauma hospital should be located based on the fact it needs to be close to a highway. Is this the same location as the cell phone tower? Why or why not? Justify your choice. 11
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Construction Graphic Organizer Type of Definition and point of Construction concurrency Construction How I constructed it Median Angle Bisector Perpendicular Bisector Altitude 13