Chapter 5 Relationships in Triangles

Section 5.1 Bisectors, Medians, and Altitudes

Perpendicular Bisector:  A line, segment or ray that intersects the midpoint of a segment at a right angle.

Theorem 5.1:  Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

Theorem 5.2:  Any point equidistant from the endpoints of the segment lies on the perpendicular bisector of the segment. (this is the converse of Theorem 5.1)

Concurrent Lines:  Three or more lines that intersect at a common point.

Point of Concurrency:  The point of intersection of concurrent lines.

 

 

Angle Bisector:  A  ray that divides an angle into two congruent angles.

Theorem 5.4:  Any point on the angle bisector is equidistant from the sides of the angle.

 

 

 

Theorem 5.5:  Any point equidistant from the sides of an angle lies on the angle bisector.  (this is the converse of Theorem 5.4)

 

Special Lines of Triangles

 

Perpendicular Bisectors of a Triangle:  run through the midpoint of a side, perpendicular to that side

Point of Concurrency of the lines is the circumcenter, which is equidistant to the vertices of the triangle. (this can be shown with a circumcircle)

 

 

Angle Bisectors:   bisects the angles of the triangle.

Point of Concurrency of the lines is the incenter, which is equidistant to the sides of the triangle. (this can be shown with an incircle)

 

 

Medians:  runs from the midpoint of a side to the opposite vertex.

Point of Concurrency of the lines is the centroid, which is the center of the surface of the triangle.

The distance from centroid to vertex is twice the distance from centroid to midpoint.

 

 

Altitudes:  segments from the vertex of the triangle in a direction perpendicular to the opposite side

            

            Point of Concurrency of the segments is the orthocenter.

 

Mid-Segments:  Segments that connect the midpoint of one side of a triangle with a midpoint of another side

Each mid-Segment is parallel to the side of the original triangle that it does not intersect. (For example FD is parallel to CB)  

 

Each mid-Segment is 1/2 the length of the side of the original triangle that it does not intersect. (For example if CB=20, then FD =___)

 

 

Find the length of each segment

 

 

Section 5.2 Inequalities and Triangles

Triangle Inequalities

The side of a triangle opposite the largest angle is the longest side.

The side of a triangle opposite the smallest angle is the shortest side.

The side of a triangle opposite the mid-sized angle is the medium length side.

List the sides in order from longest to shortest.

 

 

The angle of a triangle opposite the longest side is the largest angle.

The angle of a triangle opposite the shortest side is the smallest angle.

The angle of a triangle opposite the mid-sized side is the mid-sized angle.

List the angles in order from least to greatest.

 

List the sides in order from least to greatest

 

Exterior Angle Inequality Theorem:  An exterior angle of a triangle is greater than either of its remote interior angles.  (since the exterior angle = the sum of the two remote interiors, it must be greater than either one)

 

 

Section 5.4 The Triangle Inequality

Theorem 5.11:  The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

To simplify the concept of Th 5.11, the sum of the two shortest sides must exceed the longest side.

 

Example 1:  Can a triangle have sides of 4, 10,and 5?
 

 

 

 

Example 2:  Can a triangle have sides of 3, 7, and 5?
 

 

 

 

Find the range of possibilities for the third side of a triangle:

 

 

5 and 8

 

 

 

10 and 4

 

 

 

6 and 6