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Warm-Up Three or more lines that intersect at the same point are called concurrent lines. The point of intersection is called the point of A G C E B D F
Warm-UpThree or more lines
that intersect at the same point are called concurrent lines. The point of intersection is called the point of concurrency.
A
GC
E
B
DF
Example 1Are the lines represented by the equations
below concurrent? If so, find the point of concurrency.
x + y = 7 x + 2y = 10
x - y = 1
x=4y=3
Pick 2 equations and solve them for x & y
Plug the values into all 3 equations and see if they make true statements
Yes
5.2-5.4: Points of ConcurrencyObjectives:
1. To define various points of concurrency2. To discover, use, and prove various
theorems about points of concurrency
Intersecting Medians ActivityThe centroid of a
triangle divides each median into two parts. Click the button below to investigate the relationship of the 2 parts.
Concurrency of Medians TheoremThe medians of a
triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.
CentroidThe three medians of a
triangle are concurrent. The point of concurrency is an interior point called the centroid. It is the balancing point or center of gravity of the triangle.
Example 2In ΔRST, Q is the centroid and SQ = 8. Find QW and SW.
QW = 4SQ = 12
Others Points of ConcurrencySince a triangle has 3 sides, it seems obvious that
a triangle should have 3 perpendicular bisectors, 3 angle bisectors, and 3 altitudes. But are these various segments concurrent?
A
B
C
B
A
C
B
A
C
Others Points of ConcurrencyIn this activity, we will use patty paper to investigate
other possible points of concurrency, and then, hopefully, something magical will happen…
A
B
C
B
A
C
B
A
C
CircumcenterConcurrency of
Perpendicular Bisectors of a Triangle Theorem
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
CircumcenterThe point of concurrency of the three
perpendicular bisectors of a triangle is called the circumcenter of the triangle.
In each diagram, the circle circumscribes the triangle.
Explore Explore the perpendicular bisectors of a triangle and its circumcenter by clicking the button below
IncenterConcurrency of Angle
Bisectors of a Triangle Theorem
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
IncenterThe point of concurrency of the three angle
bisectors of a triangle is called the incenter of the triangle.
In the diagram, the circle is inscribed within the triangle.
Explore Explore the angle bisectors of a triangle and its incenter by clicking the button below
OrthocenterConcurrency of
Altitudes of a Triangle Theorem
The lines containing the altitudes of a triangle are concurrent.
G
OrthocenterThe point of concurrency of all three altitudes
of a triangle is called the orthocenter of the triangle.
The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.
Explore• Explore the altitudes of a triangle and its
orthocenter by clicking the button below.
Example 3Is it possible for any of the points of
concurrency to coincide? In other words, is there a triangle for which any of the points of concurrency are the same.
Record your thoughts/predictions in your notebook
Example 4Is it possible for any of the points of
concurrency to be collinear?
Euler LineThe Euler Line is the
line that contains the orthocenter, centroid, and the circumcenter of a triangle.
Orthocenter
Ci rcumcenter
Centroid
A
C
B
Calculate in your notebook
Calculate in your notebook
