Points of Concurrency in Triangles Notes
Circumcenter: point of concurrency of the __________________________________of the sides of a triangle.
It is equidistant from the _______________ and is the center of the ___________________circle which
is the circle around the _______________of a triangle.
The circumcenter is _____________ an acute triangle, on the ______________ of a right triangle, and is
_____________ an obtuse triangle.
http://www.mathopenref.com/trianglecircumcenter.html
Ex 1: Steps to sketch the circumcenter.
1. Draw a triangle.
2. Find the midpoint of each side. Mark the congruent parts.
3. Draw the perpendicular bisector of each side. Mark the right angles.
4. Using the circumcenter, draw a circle around the outside of the triangle.
Ex 2: Determine the value of x.
Ex 3: Graph triangle QRS with vertices Q(-6, 0), R (12, 0), and S (0, 12). Graph the perpendicular bisector of
two sides. What is the equation of each line? At what point is the circumcenter located? What is the distance
from the circumcenter to each vertex?
13
Y
12
11
10
9
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0
X
1
2
3
4
5
6
7
8
9 10 11 12 13
Points of Concurrency in Triangles Notes
Incenter: point of concurrency of the ___________________of a triangle. It is equidistant from
the __________ and is the center of the _____________circle which is the circle on the ___________of
the triangle. The incenter is always _____________ the triangle.
http://www.mathopenref.com/triangleincenter.html
Ex 3: Steps to sketch the incenter.
1. Draw a triangle.
2. Use a protractor to measure and bisect the angles. Mark the congruent angles and list the angle
measure.
3. Draw the angle bisectors. Mark the congruent angles.
4. Draw a perpendicular from the incenter to the side with a different color. Mark the right angles.
5. Using the incenter, draw a circle inside the triangle.
Ex 4: In the diagram, L is the incenter of
triangle EGJ. Determine the length of HL.
Ex 5: Determine the value of x that
makes N the incenter of the triangle.
Points of Concurrency in Triangles Notes
Centroid: point of concurrency of the _______________ of a triangle. It is _______ the
distance from the vertex to the midpoint of the opposite side. It divides the ____________into a ___:___
ratio with the longest one nearest the vertex. . It is the _____________________________________ of a
triangle, so it is always ___________ the triangle. http://www.mathopenref.com/trianglecentroid.html
Ex 6: Steps to sketch the centroid.
1. Draw a triangle.
2. Determine the midpoint of each side. Mark these appropriately.
3. Draw a line from the midpoint to the opposite vertex.
4. Measure the length from the vertex to the centroid. List the measure on the drawing.
5. Measure the length from the centroid to the side. List the measure on the drawing.
6. Write the ratio of the lengths.
Ex 7: In the figure, P is the centroid of triangle XYZ,
PY = 12, LX = 15, and LZ =18.
Ex 8: In the figure in the previous example,
if MP = 2x – 8 and MX = 3x + 3,
then what is the value of x?.
(a) What is the LY?
(b) What is NY?
(c) What is LP?
Ex 9: Graph triangle QRS with vertices Q(-6, 0), R (12, 0), and S (0, 12). Graph the median from two vertices.
What are the equations of these lines? What are the coordinates of the centroid? Show the ratio of the pieces of
Y
the centroid is 2:1.
13
12
11
10
9
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0
X
1
2
3
4
5
6
7
8
9 10 11 12 13
Points of Concurrency in Triangles Notes
Orthocenter: point of concurrency of the ______________ of a triangle. The orthocenter is _____________ an
acute triangle, the ______________ of the right angle of a right triangle, and is _____________ an obtuse
triangle.
The orthocenter, along with the _________________ and the ________________________form the Euler line.
http://www.mathopenref.com/triangleorthocenter.html
Ex 10: Steps to sketch the orthocenter
1. Draw a triangle.
2. Draw an altitude from each vertex, perpendicular to the opposite side.
Ex 11: If the legs of that meet to form vertex P of the right triangle measure 6 and 8 respectively, what is the
length of the altitude to the hypotenuse of that right triangle.
Ex 12: Graph triangle QRS with vertices Q(-6, 0), R (12, 0), and S (0, 12). Graph the altitude from two vertices.
What are the equations of these lines? What are the coordinates of the orthocenter?
13
Y
12
11
10
9
8
7
6
5
4
3
2
1
X
odd);
-7 -6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
7
8
9 10 11 12 13
P. 307 (16, 17); P. 314 (19-25
P. 322 (3-7, 33, 35); Graphing WS