Homework Solutions
Geometry CP, Dec 6
Understanding Triangle Center Concepts
Book Section: 5-5
Essential Question: What are the important triangle middle measures,
what are their characteristics, and how do I remember them?
Standards: CCSS G.CO.10, G.MG.3; G-3.5
The Basic Concepts
• Bisector – any segment, line, or plane that intersects a segment at
its midpoint.
• Perpendicular bisector – a bisector that is also perpendicular to the
segment.
• Concurrent Lines – Three or more lines that intersect at a common
point.
• Point of Concurrency – The point at which concurrent lines
intersect.
• Angle bisector – A line or plane that divides an angle into two
equal parts.
Examples
The Supporting Cast
• Median – a segment with endpoints being a vertex of a triangle
and the midpoint of the opposite side.
• Altitude – A segment from a vertex to the line containing the
opposite side perpendicular to the line containing that side. In the
formula for triangle area, altitude is known as height.
 Fact – Every triangle has three altitudes and if extended, they intersect at a
common point.
Examples
The Big Four
• Circumcenter – The point of concurrency of the
perpendicular bisectors of the sides of a triangle.
• Incenter – The point of concurrency of the angle
bisectors of a triangle.
• Centroid – The concurrency point of the medians of a
triangle.
• Orthocenter – The point of concurrency of the altitudes
of a triangle.
The Magic of the Big Four
• Circumcenter – Equidistant from the vertices of a
triangle. Can be inside, at the hypotenuse median, or
outside the triangle.
• Incenter – Equidistant from each side of the triangle by
a perpendicular segment to the side.
• Centroid – 2/3 of the total length from vertex to point,
1/3 from midpoint to point (each median).
• Orthocenter – No magic property in this one. Can be
inside, outside, or at the right angle vertex of a triangle
Examples
What You Need to Know
• Every single definition, key concept,
and characteristic of these terms, and
be able to pick them out of a crowd,
both visually and verbally.
Examples
Example to Hang Your Hat On
Name a median
Name an angle bisector
Name an altitude
Name a perpendicular bisector
Example to Hang Your Hat On
Name a median
Name an angle bisector
Name an altitude
Name a perpendicular bisector
Classwork: Quiz Retake
Homework: HW Due 12/10, Parts 1 & 2
All of it
Chapter 5 Test Next Time Also – Be
Prepared