JRLeon Discovering Geometry Chapter 3.7 HGSH

You now can perform a number of constructions in triangles, including anglebisectors, perpendicular bisectors of the sides, medians, and altitudes. In this lessonand the next lesson you will discover special properties of these lines and segments.When a set of lines has a point in common, they are concurrent. Segments, rays,and even planes are concurrent if they intersect in a single point.

Pg. 178

The point of intersection is the point of concurrency.

JRLeon Discovering Geometry Chapter 3.7 HGSH

In this investigation you will discover that some special lines in a triangle have pointsof concurrency. As a group, you should investigate each set of lines on an acute triangle, an obtuse triangle, and a right triangle to be sure that your conjectures apply to all triangles.

Draw a large triangle on patty paper. Make sure you have at least one acute triangle, one obtuse triangle, and one right triangle in your group.

Step 1

Construct the three angle bisectors for each triangle. Are they concurrent?Step 2

Compare your results with the results of others. State your observations as a conjecture.

JRLeon Discovering Geometry Chapter 3.7 HGSH

Draw a large triangle on a new piece of patty paper. Make sure you have at least one acute triangle, one obtuse triangle, and one right triangle in your group.

Step 3

Construct the perpendicular bisector for each side of the triangle and complete the conjecture.

Step 4

Draw a large triangle on a new piece of patty paper. Make sure you have at least one acute triangle, one obtuse triangle, and one right triangle in your group.

Construct the lines containing the altitudes of your triangle and complete the conjecture.

Step 5

Step 6

For what kind of triangle will the points of concurrency be the same point?Equilateral triangle

Step 7

JRLeon Discovering Geometry Chapter 3.7 HGSH

The point of concurrency for the three angle bisectors is the incenter. The Incenter of a triangle is the point where all three angle bisectors always intersect

The point of concurrency for the perpendicular bisectors is the circumcenter. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. The point of concurrency for the three altitudes is called the orthocenter. The orthocenter is the point where all three altitudes of the triangle intersect.

Use these definitions to label each patty paper from the previous investigation with the correct name for each point of concurrency. You will investigate a triangle’s medians in the next lesson.

JRLeon Discovering Geometry Chapter 3.7 HGSH

In this investigation you will discover special properties of the circumcenter.

Using your patty paper from Steps 3 and 4 of the previous investigation, measure and compare the distances from the circumcenter to each of the three vertices. Are they the same? Compare the distances from the circumcenter to each of the three sides. Are they the same?

Tape or glue your patty paper firmly on a piece of regular paper. Use a compass to construct a circle with the circumcenter as the center and that passes through any one of the triangle’s vertices.What do you notice?

Use your observations to state your next conjecture.

Step 1

Step 2

Step 3

JRLeon Discovering Geometry Chapter 3.7 HGSH

In this investigation you will discover special properties of the incenter.

Using the patty paper from the first two steps of Investigation 1, measure and compare the distances from the incenter to each of the three sides. (Remember to use the perpendicular distance.) Are they the same?

Construct the perpendicular from the incenter to any one of the sides of the triangle. Mark the point of intersection between the perpendicular line and the side of the triangle.

Tape or glue your patty paper firmly on a piece of regular paper. Use a compass to construct a circle with the incenter as the center and that passes through the point of intersection in Step 2.What do you notice?

Step 4

Step 3

Step 2

Step 1

Use your observations to state your next conjecture.

JRLeon Discovering Geometry Chapter 3.7 HGSH

You just discovered a very useful property of the circumcenter and a very useful property of the incenter. You will see some applications of these properties in the exercises. With earlier conjectures and logical reasoning, you can explain why your conjectures are true.

JRLeon Discovering Geometry Chapter 3.7 HGSH

JRLeon Discovering Geometry Chapter 3.7 HGSH

Classwork/Homework: Lesson 3.7 Pages 181-182 Problems 1 thru 8

JRLeon Discovering Geometry Chapter 3.7 HGSH

JRLeon Discovering Geometry Chapter 3.7 HGSH

JRLeon Discovering Geometry Chapter 3.7 HGSH