Using Algebra to find points of Concurrency
Name ___________________________________ Date ___________
You have been using constructions to find the points of concurrency
of triangles. You can identify these points (ordered pair) by using algebra.
We will do this first one together, and then you can do the one on the back.
Given VABC with vertices A ( 0, −4) , B ( −4, 4 ) , and C (8,8). (sketch on the grid to the right)
Complete the following table using the above information
Side
Slope
⊥ slope
midpoint
AB
BC
AC
Complete the following:
• The circumcenter is the intersection of the _______________________________.
•
The incenter is the intersection of the ___________________________________.
•
The orthocenter is the intersection of the ________________________________.
•
The centroid is the intersection of the ___________________________________.
•
On Euler’s segment you will find _______________________________________.
Using the information in your table, algebraically find the coordinates of the circumcenter.
( ________, _________ )
Using the information in your table, algebraically find the coordinates of the orthocenter.
( ________, _________ )
Using the information in your table, algebraically find the coordinates of the centroid.
( ________, _________ )
Now, using two of the above, find the equation of Euler’s line for ΔABC .
y = _________________________
Repeat the process for the following triangle:
ΔJIM with vertices J 1,7 , I −3,2 , and M 7,−6 .
( ) (
)
(
)
(sketch the triangle to the right – this will help you to do an estimated check of your work)
Side
Slope
⊥ slope
midpoint
JI
IM
JM
Give the ordered pair that identifies the following:
Circumcenter _(____________)__ Orthocenter _(____________)_ Centroid _(__________)_
Equation of Euler’s line
y = ________________________