GeometryUnit 5: Triangle Parts
Concurrent:
When three or more lines intersect at the same point, P
P
Triangle Midsegment:
Segment connecting the midpoints of two sides of a triangle
Triangle Midsegments
• Parallel to the third side of the triangle
Triangle Midsegments
• Parallel to the third side of the triangle
• Half the length of the third side of the triangle
Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.
Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.
• The circles will be inscribed in or circumscribed about triangles.
PERPENDICULAR BISECTORS
Perpendicular Bisectors
• Lines that bisect a side and are perpendicular to it
PERPENDICULAR BISECTOR
Bisects a side and makes a 90 angle with it
Perpendicular Bisectors (in purple)
• Concurrent at the CIRCUMCENTER of each green triangle
Perpendicular Bisectors
• Circumcenter can be inside, on, or outside of the triangle
Perpendicular Bisectors
• Circle 1: circumcenter is outside triangle• Circle 2: circumcenter is inside triangle
. .circumcenter
circumcenter
Perpendicular Bisectors
• purple radii of circle go from circumcenter to each vertex of the triangle
r
r
Perpendicular Bisectors
• Can you identify three isosceles triangles in each figure?
r
r
Perpendicular Bisectors
• Lines that bisect a side and are perpendicular to it
• Concurrent at the Circumcenter of the triangle
• Circumcenter can be inside, on, or outside of the triangle
• Radii of circle go from circumcenter to each vertex of the triangle
Angle Bisector
Bisects the angle
Angle BisectorConcurrent at the INCENTER
(center of circle INSCRIBED in triangle)
Angle BisectorConcurrent at the INCENTER
(center of circle INSCRIBED in triangle)
MEDIANSegment from vertex to opposite
side’s midpoint
(Nothing to do with the angle!!)
MEDIAN
Concurrent at CENTROID
(center of gravity)
MEDIAN
The center of gravity is the BALANCE POINT.
MEDIAN
The CENTROID must be inside of the triangle!
ALTITUDE
Perpendicular segment from a vertex to the side opposite (or extension of the side
opposite).
ALTITUDE
Height of the triangle
(perpendicular to the base)
ALTITUDES
Concurrent at the ORTHOCENTER
orthocenter.
ALTITUDES
Can be outside of a triangle.
altitude
MEDIAN
Concurrent at CENTROID
(center of gravity)
. Centroid
Angle BisectorConcurrent at the INCENTER
(center of circle INSCRIBED in triangle)
Angle BisectorConcurrent at the INCENTER
(center of circle INSCRIBED in triangle)
. Incenter
ALTITUDES
Concurrent at the ORTHOCENTER
orthocenter.
PERPENDICULAR BISECTORS
Perpendicular Bisectors:
Concurrent at the Circumcenter
. circumcenter
Perpendicular Bisectors
• Radii of circle go from circumcenter to each vertex of the triangle
r
r
r
Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.
Problem: Three cities want to build a park that is the same distance from each of their city centers. What should they do?
Kenmore
Shoreline
MLT
Which “triangle center” will be the same distance from each city center?
Shoreline
Kenmore
Shoreline
MLT
The CIRCUMCENTER
Shoreline
Kenmore
Shoreline
MLT
Which triangle segments or lines are used to find the circumcenter?
Shoreline
Kenmore
Shoreline
MLT
PERPENDICULAR BISECTORS
Shoreline
Kenmore
Shoreline
MLT
PERPENDICULAR BISECTORS(green lines) are concurrent at the
CIRCUMCENTER.
Kenmore
MLT
CircumcenterShoreline
The Circumcenter is equidistant from each city center.
Kenmore
MLT
Circumcenter
Shoreline
The distance is the RADIUS of the circle centered at the CIRCUMCENTER.
Kenmore
MLT
C
Shoreline
r
r r
Problem: Three cities want to build a toxic waste dump that is the same distance from each of their city centers. What should they do?
MLTKenmore
Shoreline
Which “triangle center” will be the same distance from each city center?
MLTKenmore
Shoreline
Which “triangle center” will be the same distance from each city center? The CIRCUMCENTER
MLTKenmore
Shoreline
Which triangle segments or lines are used to find the circumcenter?
MLTKenmore
Shoreline
Which triangle segments or lines are used to find the circumcenter?
PERPENDICULAR BISECTORS
MLTKenmore
Shoreline
PERPENDICULAR BISECTORSare concurrent at the CIRCUMCENTER.
Kenmore
Shoreline
MLT
Circumcenter
The Circumcenter is equidistant from each city center.
KenmoreShoreline
MLT
Circumcenter
The distance is the RADIUS of the circle centered at the CIRCUMCENTER.
KenmoreShoreline
MLT
Circumcenter
radius
The circumcenter can be outside of the triangle.
KenmoreShoreline
MLT
Circumcenter
radius
The centroid and incenter must be inside
of the triangle.