The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + vector OC)/3.
A. Use vector techniques to prove that a triangle and its medial triangle have the same centroid, stating each step of the proof.
1. Provide written justification for each step of your proof.
B. Provide a convincing argument short of a proof (suggested length of 3-4 sentences) that the theorem is true.© BrainMass Inc. brainmass.com October 25, 2018, 9:44 am ad1c9bdddf
A. Let us assume mid points of AB, BC and CA are L, M and N respectively.
Let us assume centroid of ABC is G, and of LMN is G'.
vector(OG) = (vector(OA) + vector(OB) + vector(OC))/3
L is mid point of AB
vector(OL) = vector(OA) + vector(AL)
As AL = LB = AB/2
vector(OL) = vector(OA) + vector(AB/2)
=> vector(OL) = ...
This solution provides proof that the centroid of a triangle and the centroid of medial triangle coincide.
Vertex Figures and Tessellations
Another definition of a regular tessellation is one whose vertex figures are identical regular polygons.
A vertex figure is made by connecting the midpoints of all the edges which touch a given vertex.
(1) Sketch the vertex figures for the regular semiregular tessellation of the plane and verify the definition.
(2) The dual of the tessellation is a new tessellation formed by connecting the centers (centroids) of polygons that share a common side. Find the duals of the regular tessellation.
(3) Common notation for describing regular or semiregular tessellations uses vertex arrangements, which list the number of sides of each polygon going around the vertex. Give the vertex arrangements for the regular and semiregular tessellations.View Full Posting Details