The Centroid

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The Centroidand some properties

• Armando Martinez-Cruz
• Amartinez-cruz_at_fullerton.edu
• in consultation with
• Coach Karen Delaney
• Mathcoach_at_adelphia.net
• Buena Park Cluster
• June 25, 2005
• TASEL-M

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Mathematics and Pedagogy

• Proof
• Mathematical Reasoning
• Geometric Constructions
• Folding patty paper

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Comments on learning difficulties (next four
slides)

• Learners have difficulties recognizing triangles
  with the same area, especially when the triangles
  are completely different. The reasoning in this
  activity is based totally in this idea (triangles
  with the same area). Therefore, it is important
  that time is spent on this part before moving on.
  

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• In the next figure, let points C and D lie on a
  parallel to segment AB. Then, triangles ABC and
  ABD have the same area.
• Reason They both have the same base (AB) and
  height (the distance from the parallel to the
  segment is unique).

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• Conversely, let A, B, C and D be four points on a
  line L. Let AB be congruent to CD. If E is a
  point not on the line L, the triangles ABE and
  CDE have the same area. Again, both triangles
  have the same base (since AB and CD are congruent
  segments) and the distance from a point to a line
  is unique.

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The Medians in a Triangle

• A median in a triangle ABC is the segment that
  joints a vertex with the midpoint of the opposite
  side.
• Hence there are three medians in a triangle.

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Medians are concurrent

• The medians in a triangle are concurrent (i.e.,
  they meet in one interior point of the triangle.)
  
• The point of concurrency is the centroid of a
  triangle.

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Geometric Constructions folding (patty) paper

• At this point audience will use patty paper to
  determine the midpoint of a segment, the medians
  of a triangle, and their intersection point (the
  centroid).

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• The centroid has many properties
• It is the center of gravity of the triangle
• It splits the triangle in six small triangles
  with the same area
• It splits the medians in the ratio 12

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Center of Gravity

• The centroid is the center of gravity of the
  triangle (i.e., if the triangle is a sheet of
  iron and we want the triangle to be in balance
  over a sharp pin, we have to place the triangle,
  in such a way that the centroid is over the pin).

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The area of all six little triangles is the
SAME!

• The centroid splits the triangle in six small
  little triangles.
• All of them with the same area!!!

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• For instance, the blue and the yellow triangles
  have the same area since both bases are congruent
  (remember that Mc is a midpoint) and the distance
  from the centroid to AB is the common--height
  for both triangles.

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• Notice that triangles CAMc and CMcB have the same
  area as well (again the base is the same and the
  height is the same).

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• So the orange triangle and the pink triangle have
  the same area (since we already proved that the
  yellow triangle and the blue triangle have the
  same area). And this in turn shows that all six
  little triangles have the same area.

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The centroid splits each median into two
segments, one being half the length of the other.

• From all previous work, triangle CAG is twice the
  area of triangle CGMa.
• And both have the same height! Lets call it h.
• Let B be the base of CAG and b the base of CGMa.
  So the areas are related as 1/2Bh 2(1/2)bh,
  which says that B 2b as we wanted.