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Symbol Defintion
Sqr(x) = Square root of x
GE 14 01. What is centroid
Definition
- Medians of triangle are concurrent at a point G which is called centroid
- Centroid is also called gravity center of a triangle
- Three lines meet at one point is called concurrent
- Vertex of triangle to mid point of opposite side is called median
- AD, BE, CF are medians of triangle as shown in diagram
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GE 14 02. Medians of triangle are concurrent
diagram
- Draw triangle ABC
- Let D be mid point of BC, E be midpoint of CA
- Draw medians AD and BE intersecting at G
- Join C and G. Produce CG to F on AB
- Produce CF to H and let FH = CG
- We want to prove that F is mid point of AB
- In triangle CAH E and G are mid points of CA and CH
- Hence EG parallel to AH (mid point theory)
- In triangle CBH D and G are mid points of BC and CH
- Hence DG parallel to BH
- Hence AGBH is a parallelogram (opposite sides parallel)
- Hence AF = BF (properties of parallelogram)
- Hence F is mid point and CF is medain passing G
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GE 14 03. Centroid to vertex is 2/3 of median
- In Q02, FH = FG (properties of parallelogram)
- Hence FG = GH/2 = CG/2
- CF = CG + FG = CG + CG/2 = 3*CG/2
- Or CG = 2*CG/3 where CG is the median
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Q04. Locus of centroid
diagram
- Let A and B are fixed and C is moving with angle ACB = constant
- Find locus of centroid
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Diagram of geometry
Program 10 03
- Draw GP parallel to CA and P on AB
- Draw GQ parallel to BC and Q on AB
- Hence angle PGQ = angle ACB (fixed)
- Since GP parallel to CA hence AP = 2*AF/3
- Since GQ parallel to BC hence BQ = 2*BF/3
- AB is fixed and then AF and BF are fixed
- Hence P and Q are fixed points.
- Also PGQ is fixed angle.
- Hence locus of G is an arc of circle passing P, G, Q.
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GE 14 05. Coordinate geometry Prove that medians of triangle concurrent
Reference
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Coordinate geometry
Program 11 05
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GE 14 06. Construct triangle if 3 medians are given
- See GE 03 12
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GE 14 07. Answer
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GE 14 08. Answer
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GE 14 09. Answer
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GE 14 10. Answer
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GE 14 00. Outline
- Centroid : medians are concurrent
- Two third rule : Centroid to vertex is 2/3 of it median
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