Bisectors, Medians, and Altitudes

Use Medians and Altitudes

Geometry Honors

Ms. Hogan

Recall: Perpendicular Bisectors

A segment, line, ray, or plane that is perpendicular to a segment at its midpoint.

Recall: Perpendicular Bisectors

Circumcenter (Point of Concurrency of Perpendicular Bisectors): The point where three perpendicular bisectors of a triangle intersect. (Does not necessarily have to be inside the triangle.)

Recall: Perpendicular Bisectors

The circumcenter, L, is equidistant from all three vertices of the triangle.

Recall: Angle Bisectors of Triangles

An angle bisector of a triangle is a segment, ray, line, or plane that divides an angle into two congruent parts, called adjacent angles.

Recall: Angle Bisectors of Triangles

Incenter (Point of concurrency of angle bisectors): The point where all three angle bisectors intersect.

Recall: Angle Bisectors of Triangles

The incenter, I, is equidistant from all three sides of the triangle. The incenter is the center of a circle inscribes within the triangle that touches all three sides.

Multiple Choice

Name the dashed segment:

• altitude
• angle bisector
• perpendicular bisector
• median

Multiple Choice

The figure is an example of a(n) ...

• altitude 
• perpendicular bisector
• midsegment
• angle bisector

Multiple Choice

Which point of concurrency is equidistant from the three vertices of a triangle?

• Circumcenter

• Centroid

• Orthocenter

• Incenter

Multiple Choice

Which point of concurrency is equidistant from the three sides of a triangle?

• Incenter

• Centroid

• Circumcenter

• Orthocenter

Multiple Choice

Which point of concurrency is the intersection of the perpendicular bisectors of the triangle?

• Centroid

• Incenter

• Orthocenter

• Circumcenter

Multiple Choice

Which point of concurrency is the intersection of the angle bisectors of a triangle?

• Centroid

• Incenter

• Orthocenter

• Circumcenter

Multiple Choice

The circumcenter is located outside the triangle.

• Sometimes

• Always

• Never

Multiple Choice

The incenter is located outside the triangle.

• Sometimes

• Always

• Never

Multiple Choice

QR is a midsegment of triangle WUV. What is the length of segment QR?

• 4

• 8

• 16

• 32

New News: Median

A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

New News: Centroid

The point of concurrency of the medians is called the Centroid.

New News: Centroid

Theorem: The centroid is 2/3 (two thirds) of the distance from each vertex to the midpoint of the opposite side.

Example:

Find the length of the median.

Example:

There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P.

Note that PC is the segment containing the vertex, so it is the segment that is 2/3 the length of the whole segment.

Multiple Choice

Fill in the blank.
All three medians intersect at the point of concurrency called the ____________.

• Incenter
• Circumcenter
• Orthocenter
• Centroid

Multiple Choice

Fill in the blanks.
Medians connect the __________ to the opposite ________.

• hypotenuse; leg
• leg; hypotenuse
• vertex; leg
• leg; vertex

Multiple Choice

In ΔXYZ, YB, XC, and ZA are medians, and YC = 15.

Find the length of YZ.

• 30

• 15

• 5

• 7.5

Multiple Choice

Given segment AM is a median in ΔABC. Find the length of segment BC.

• 34

• 6

• 17

• 27

New News: Altitude

An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

New News: Orthocenter

The point of concurrency of the three altitudes of a triangle is called the orthocenter.

Final note:

In an isosceles triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment.

In an equilateral triangle, this is true for the special segment from any vertex.

Multiple Choice

The altitudes of a triangle intersect at the _________________________.

• Centriod
• Circumcenter
• Incenter
• Orthocenter

Multiple Choice

Fill in the blank.
Altitudes hit the ground at _______ degrees.

• 180
• 90
• 60
• 360

Great Job!

More information and practice can be found in your HMH Textbook L8.3 Median and Altitude & L8.4 Midsegment