Concept Byte: Special Segments in Triangles

Use With Lesson 5-4

TECHNOLOGY

You already know about two sets of lines that are concurrent for any triangle. In the following activity, you will use geometry software to confirm what you know about the concurrency of a triangle's perpendicular bisectors and angle bisectors. Then you will explore two more sets of special segments in triangles.

Activity

Use geometry software.

• Construct a triangle and the three perpendicular bisectors of its sides. Use your result to confirm Theorem 5-6, the Concurrency of Perpendicular Bisectors Theorem.
• Construct a triangle and its three angle bisectors. Use your result to confirm Theorem 5-7, the Concurrency of Angle Bisectors Theorem.

• An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Construct a triangle. Through a vertex of the triangle construct a segment that is perpendicular to the line containing the side opposite that vertex. Next construct the altitudes from the other two vertices.

  

• A median of a triangle is the segment joining the midpoint of a side and the opposite vertex. Construct a triangle. Construct the midpoint of one side. Draw the median. Then construct the other two medians.

  

Exercises

1. What property do the lines containing altitudes and the medians seem to have?

   Does the property still hold as you manipulate the triangles?

2. State your conjectures about the lines containing altitudes and about the medians of a triangle.
3. Copy the table. Think about acute, right, and obtuse triangles. Use inside, on, or outside to describe the location of each point of concurrency.

   	Perpendicular Bisectors	Angle Bisectors	Lines Containing the Altitudes	Medians
   Acute Triangle	begin box , , end box	begin box , , end box	begin box , , end box	begin box , , end box
   Right Triangle	begin box , , end box	begin box , , end box	begin box , , end box	begin box , , end box
   Obtuse Triangle	begin box , , end box	begin box , , end box	begin box , , end box	begin box , , end box

4. Extend What observations, if any, can you make about these special segments for isosceles triangles? For equilateral triangles?

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