Assignment P. 307-309: 16, 17, 25, 28a, 30 P. 314-315: 19-25, 29 P. 322-323: 3, 5, 7, 8, 16, 33, 35,...
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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
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Transcript of Assignment P. 307-309: 16, 17, 25, 28a, 30 P. 314-315: 19-25, 29 P. 322-323: 3, 5, 7, 8, 16, 33, 35,...
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 concurrency.
A
GC
E
B
D
F
Example 1
Are 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 Concurrency
Objectives:
1. To define various points of concurrency
2. To discover, use, and prove various theorems about points of concurrency
Intersecting Medians Activity
The 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 Theorem
The 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.
Centroid
The 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 2
In 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
Circumcenter
Concurrency 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.
Circumcenter
The 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
Incenter
Concurrency 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.
Incenter
The 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
Orthocenter
Concurrency of Altitudes of a Triangle Theorem
The lines containing the altitudes of a triangle are concurrent.
G
Orthocenter
The 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 3
Is 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 4
Is it possible for any of the points of concurrency to be collinear?
Euler Line
The Euler Line is the line that contains the orthocenter, centroid, and the circumcenter of a triangle.
Orthocenter
Circumcenter
Centroid
A
C
B
Calculate in your notebook
Calculate in your notebook
