Constructing Points of Concurrency. Geometry. Chapter 3.7 Constructing Points of Concurrency. Objectives: 1. Discover points of concurrency of the angle bisectors, perpendicular bisectors. 2. Explore the relationships between points of concurrency and inscribed and circumscribed circles. 3. Learn new terms. HW : Lesson 3.7 pg.181-182. # 1,2,3,4,6,7

1. Do now. 2. Sketch distances from point E to lines AB and BD 3. What do you know about those distances?

What does Concurrent mean? Points of Concurrency What does Concurrent mean?

Construct: Angle Bisector Points of Concurrency Construct: Angle Bisector ✔ Incenter (Angle Bisectors) A B1 C C1 A1 B Incenter

The incenter is the center of the triangle's inscribed circle!!!

Construct: Perpendicular Bisector Points of Concurrency Construct: Perpendicular Bisector ✔ Circumcenter (Perpendicular Bisectors) A b1 c1 C a1 Circumcenter B

The circumcenter is the center of the triangle's circumscribed circle!!!

Practice Book: 1. A circular revolving sprinkler needs to be set up to water every part of a triangular garden. Where should the sprinkler be located so that it reaches all of the garden, but doesn’t spray farther than necessary?

2. You need to supply electric power to three transformers, one on each of three roads enclosing a large triangular tract of land. Each transformer should be the same distance from the power-generation plant and as close to the plant as possible. Where should you build the power plant, and where should you locate each transformer?

Constructing Points of Concurrency. Geometry. Chapter 3.7 Constructing Points of Concurrency. Objectives: 1. Points of concurrency of the angle bisectors, perpendicular bisectors, and altitudes of a triangle. 2. Explore the relationships between points of concurrency and inscribed and circumscribed circles. 3. Learn new terms. HW : Lesson 3.7 pg.183-184 # 12, 22-26 Do now: a) Construct 60˚ angle b) Construct 30 ˚ angle c) Construct 45 ˚ angle

Construct: Drop a perpendicular. Points of Concurrency Construct: Drop a perpendicular. Orthocenter (Altitudes) ✔ B1 A C C1 A1 B Orthocenter

Conjectures Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle __________ _________________________. Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of a triangle __________________. Altitude Concurrency Conjecture The three altitudes (or lines containing the altitudes) of a triangle _________________. meet at a point (are concurrent) are concurrent are concurrent

More Conjectures Circumcenter Conjecture The circumcenter of a triangle __________________ ____________________. Incenter Conjecture The incenter of a triangle ________________________ ___________. is equidistant from the vertices is equidistant from the sides

Vocabulary E ____ Concurrent ____ Point of Concurrency D ____ Incenter A) The point of concurrency for the three angle bisectors is the incenter. B) The point of concurrency for the three altitudes. C) The point of concurrency for the perpendicular bisector. D) The point of intersection E) Three or more lines have a point in common. E ____ Concurrent ____ Point of Concurrency ____ Incenter ____ Circumcenter ____ Orthocenter D A C B

3. Draw an obtuse triangle 3. Draw an obtuse triangle. Construct the inscribed and the circumscribed circles.

4. Construct an equilateral triangle 4. Construct an equilateral triangle. Construct the inscribed and the circumscribed circles. How does this construction differ from Exercise 3?

5. Construct two obtuse, two acute, and two right triangles 5. Construct two obtuse, two acute, and two right triangles. Locate the circumcenter of each triangle. Make a conjecture about the relationship between the location of the circumcenter and the measure of the angles.