CC Geometry H
Aim #14: How do we prove using coordinate geometry that the three medians of
any triangle are concurrent at a point called the centroid? How do we write
equations of perpendicular bisectors?
Do Now: Using the triangles below, sketch median AB from vertex A. Then,
define a median of a triangle in your own words.
A
A
A
Opening Exercise:
Let A(30,40), B(60,50) and C(75,120) be the vertices of a triangle.
A) Find the coordinates of the midpoint M of AB. Then, find the point G1 that is
the point one-third of the way along MC, closer to M than to C.
B) Find the coordinates of the midpoint N of BC. Then, find the point G2 that is
the point one-third of the way along NA, closer to N than to A.
C) Find the coordinates of the midpoint R of CA. Then, find the point G3 that is
the point one-third of the way along RB, closer to R than to B.
d) What are the coordinates of G1, G2, and G3? _________________
e) G1, G2, and G3 represent the point at which the three medians intersect. This
point of concurrency is called the ______________ and it divides each of the
three medians such that this point is ______ the distance from the side of the
triangle to the opposite vertex. This point divides the median into a ratio of ____.
Exercise 2: The class will be divided into thirds and each group will
complete either part A, B, or C. We will prove that the three medians of ANY
triangle are concurrent at the centroid.
Given: A(a1,a2), B(b1,b2) and C(c1,c2) are the vertices of a triangle. Find the
coordinates of the point of concurrency of the three medians.
A) Find the coordinates of the midpoint M of AB. Then, find the point G1 that is
the point one-third of the way along MC, closer to M than to C.
B) Find the coordinates of the midpoint N of BC. Then, find the point G2 that is
the point one-third of the way along NA, closer to N than to A.
C) Find the coordinates of the midpoint R of CA. Then, find the point G3 that is
the point one-third of the way along RB, closer to R than to B.
d) Let A(-23,12), B(13,36), and C(23,-1) be vertices of a triangle. Where will the
medians of this triangle intersect (what are the coordinates of the centroid)?
3) Find the coordinates of the intersection of the medians of ΔABC given A(3,3),
B(7,9) and C(11,3).
4) Find the coordinates of the centroid for each graph:
a)
b)
Writing the equation of the perpendicular bisector of a segment:
1) Write an equation of the line that is the
perpendicular bisector of the line segment
having endpoints (3,-1) and (3,5).
2) Line A is the perpendicular bisector of BC with B(-2,-1) and C(4,1).
a. What is the midpoint of BC?
b. What is the slope of BC? of line A?
c. What is the equation of line A,
the perpendicular bisector of BC?
3) Write an equation of the perpendicular bisector of the line segment whose
endpoints are (-1,1) and (7,-5).
4) GDAY is a rhombus. If point G has coordinates (2,6) and A has coordinates
(8,10), what is the equation of the line that contains the diagonal DY of the
rhombus?
D
G
A
Y
Name: ____________________
CC Geometry H
Date: ___________
HW #14
1) Find the coordinates of the intersection of the medians of ΔABC given A(-5,3),
B(6,-4) and C(10,10).
2) Let A(0,7), B(-6,1), and C(6,-5) be vertices of a triangle. Where will the medians
of this triangle intersect?
3) Two vertices of a triangle are (0, 0) and (9, 0). The centroid is (6, 1). Find the
third vertex of the triangle.
4) Determine the distance between point A(-1,-3) and B(5,5). Then, write the
equation of the perpendicular bisector of AB.
5) The coordinates of the endpoints of AB are A(0,0) and B(0,6). Write the
equation of the perpendicular bisector of AB.
6) The coordinates of 2 vertices of square ABCD are A(2,1) and B(4,4). Determine
the slope of side BC. Then, write the equation of the line that is parallel to BC and
goes through the midpoint of AB.
B
A
Mixed Review:
1) With a compass and straigtedge, construct the median from vertex C in the
triangle below. Leave all construction marks.
A
B
C
2) Triangle ABC with vertices A(4,-1), B(5,6), and C(1,3). Use coordinate geometry
to prove that ABC is an isosceles right triangle. Write a concluding statement.
3) Find the point on the directed line segment from (0,3) to (6,9) that divides the
segment in the ratio of 2:1.