Geometry. Chapter 3.7Constructing 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?

Points of ConcurrencyWhat does Concurrent mean?

✔ Incenter (Angle Bisectors)

Construct: Angle Bisector

A

B

C

A1

B1

C1

Points of Concurrency

Incenter

The incenter is the center of

the triangle's inscribed circle!!!

Circumcenter (Perpendicular Bisectors)

A

B

C c1

a1

b1

✔

Points of ConcurrencyConstruct: Perpendicular Bisector

Circumcenter

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 thepower plant, and where should you locate each transformer?

Geometry. Chapter 3.7Constructing 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

Orthocenter (Altitudes)

A

B

C

C1

A1

B1

✔

Points of ConcurrencyConstruct: Drop a perpendicular.

Orthocenter

ConjecturesAngle Bisector Concurrency ConjectureThe three angle bisectors of a triangle __________ _________________________.

Perpendicular Bisector Concurrency ConjectureThe three perpendicular bisectors of a triangle __________________.

Altitude Concurrency ConjectureThe three altitudes (or lines containing the altitudes) of a triangle _________________.

meet at a point (are concurrent)

are concurrent

are concurrent

Circumcenter ConjectureThe circumcenter of a triangle __________________ ____________________.

Incenter ConjectureThe incenter of a triangle ________________________ ___________.

is equidistant from the vertices

is equidistant from the sides

More Conjectures

____ Concurrent

____ Point of Concurrency

____ Incenter

____ Circumcenter

____ Orthocenter

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

D

A

C

B

Vocabulary

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

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. Locate the circumcenter of each triangle. Make a conjecture about the

relationship between the location of the circumcenter and the measure of the angles.