Special Relationships in Triangles

Michal Grissom, Sep 16, 2019

Student paced lesson determining special parts of triangles

Table of Contents

  1. Medians
    1. Median in Triangles
    2. Special Properties of the Centroid
  2. Perpendicular Bisectors
    1. Perpendicular Bisectors in Triangles
    2. What does Circumcenter mean?
  3. Angle Bisectors
    1. Angle Bisectors in Triangles
    2. What does Incenter mean?
  4. Altitudes
    1. Altitudes in Triangles
  5. Eulers Line
    1. Constructing Eulers Line
    2. Special Triangles

1.

Medians

  • 1. Median in Triangles
  • 2. Special Properties of the Centroid

1.1.

Median in Triangles

Finding the Medians
1) Find and Label the midpoint of each side.[br]2) Connect the midpoint of each side to it's OPPOSITE vertex[br]3) What do you notice about all 3 medians?
Medians
This point in which each median intersects is called the Centroid
The Centroid is called the balancing point of the triangle. If you were to place your finger directly underneath the centroid, it will balance perfectly since it is the "center of gravity". [br]Now move point A so that it is an Obtuse Triangle, and then a Right Triangle. What do you notice about the centroid? How does it move?
Will the Centroid always be inside the circle?
– Michal Grissom

1.2.

–

2.

Perpendicular Bisectors

  • 1. Perpendicular Bisectors in Triangles
  • 2. What does Circumcenter mean?

2.1.

Perpendicular Bisectors in Triangles

1) Find and label the midpoint of each side of the triangle. [br]2) Using the Perpendicular line tool, construct a line perpendicular to BC that goes through its midpoint.[br]3) Repeat this process for the other sides of the triangle. [br]4) What do you notice about each of the 3 perpendicular bisectors?
This point of concurrency is called the Circumcenter
Find the distance from the circumcenter to each of the vertices. What do you notice?[br]Now, Move point A so that you have created an obtuse triangle, and then a right triangle. What happens to the circumcenter as the triangle changes?
If the triangle is obtuse, where is the circumcenter located?
– Michal Grissom

2.2.

–

3.

Angle Bisectors

  • 1. Angle Bisectors in Triangles
  • 2. What does Incenter mean?

3.1.

Angle Bisectors in Triangles

1) Use your geogebra tools to create an angle bisector for angle A.[br]2) Repeat this process for angle B, and C. [br]3) What do you notice about all 3 angle bisectors?
This point of concurrency is called the Incenter.
Use our distance tool to find the distance from the incenter to each side of the triangle. What do you notice? Now, move point A so that the triangle is obtuse, and then again to make it a right triangle. What happens to the incenter as the triangle changes?
Where is the incenter always located?
– Michal Grissom

3.2.

–

4.

Altitudes

  • 1. Altitudes in Triangles

4.1.

Altitudes in Triangles

Finding an Altitude
1) Find the Perpendicular line tool[br]2) Click on a side of the triangle and then click on it's OPPOSITE vertex. [br]3) Perform this task on each of the 3 sides of the triangle. [br]4) What do you notice about all 3 Altitudes?
Altitudes
This point at which each altitude intersects is called the Orthocenter
Now move point A so that it is an Obtuse Triangle, and then a Right Triangle. What do you notice about the Orthocenter? How does it move?
Where is the Orthocenter located on a right triangle?
– Michal Grissom

5.

Eulers Line

  • 1. Constructing Eulers Line
  • 2. Special Triangles

5.1.

Constructing Eulers Line

Use the checkboxes to show the Orthocenter, Incenter, Circumcenter, and Centroid.
Can you create a line that goes through at least 3, if not all of the special points in this triangle?[br]This line is called Euler's Line. [br]Move vertex A to make the triangle Obtuse and then Right. What do you notice about the line? How does it move? What special points does it go through?
What point does Euler's line not necessarily have to go though?
– Michal Grissom

5.2.

–

Information