Constructionist 1
Centroid & Orthocenter
Independent Work
1. What is meant by the term “concurrency”?
2. Define the following terms with reference to a triangle:
a. Median:
b. Altitude:
3. Describe (in words) how to construct the centroid of a triangle. Include a diagram in your
response.
4. Describe (in words) how to construct the orthocenter of a triangle. Include a diagram in
your response.
Created by K. Schmarge, 2014
Constructionist 1
Centroid & Orthocenter
Independent Work
5. In the first link (Geogebra), you explore interactive diagrams. Consider the one labeled
Centroid and answer the questions to the right of the diagram. The questions have been
reprinted here:
A. Is it possible to have the centroid OUTSIDE the triangle?
B. Where is the centroid located in an ACUTE triangle?
C. Where is the centroid located in an OBTUSE triangle?
D. Where is the centroid located in a RIGHT triangle?
E. Can the centroid ever be closer to the vertex of a triangle than it is to the opposite
side?
6. On the same page referred to in the previous problem, consider the interactive diagram
labeled Orthocenter and answer the questions on its right. The questions have been
reprinted here:
A. Is it possible to have the orthocenter OUTSIDE the triangle?
B. Where is the orthocenter located in an ACUTE triangle?
C. Where is the orthocenter located in an OBTUSE triangle?
D. Where is the orthocenter located in a RIGHT triangle?
E. If you made a triangle using A, B, and the orthocenter, what would be the
orthocenter of THAT triangle?
Created by K. Schmarge, 2014
Constructionist 1
Centroid & Orthocenter
Independent Work
7. Explain the geometric properties and significance of the centroid and orthocenter. You
may find the Math Open Reference link to be most useful for this.
8. The median of a triangle divides a triangle into two triangles of equal area. Demonstrate
why this is true in a paragraph proof. Does this mean that the two triangles created must
be congruent? Explain.
Created by K. Schmarge, 2014
Constructionist 1
Centroid & Orthocenter
Independent Work
9. In the diagram shown at right, T is the centroid. Find each of the following values, treating
each lettered problem separately.
A
A. If AF = 4, then FB = ______
B. If FT = 5, then TC = ______
F
E
C. If AT = 9, then TD = ______
D. If BE = 24, then TE = ______ and BT = ______
E. If AT = 36, then AD = ______
T
B
D
C
F. If BC = 36, then DC = ______
G. If TD = 4x + 5 and AT = 9x, find x = ______ and AD = ______
H. If FT = 2x – 8 and FC = 3x + 3, find x = ______ and TC = ______
I. If BT = 5x and TE = 3x – 2, find x = ______ and BE = ______
10. What is the Euler Line? How do the centroid and orthocenter relate to the Euler Line?
Explain.
Created by K. Schmarge, 2014
Constructionist 1
Centroid & Orthocenter
Independent Work
11. Graphing Exercise
A. Graph the points A(–1, 2), B(5, 6), C(5, –2)
and find the coordinates of the
of the centroid.
B.Graph the points A(0, 4), B(3, 10),
&C(6, –2) and find the coordinates
of the orthocenter.
FORMULAS: If necessary, look-up midpoint formula, slope formula, perpendicular line slopes,
and slope-intercept form of a line.
Created by K. Schmarge, 2014