Mathematics Dictionary
Dr. K. G. Shih
Centroid : Gravity cneter
Subjects
Symbol Defintion
Sqr(x) = Square root of x
Q01 |
- What is centroid ?
Q02 |
- Prove that medians of triangle are concurrent
Q03 |
- Centroid to vetex is 2/3 of it median
Q04 |
- Locus of centroid
Q05 |
- Using coordinate geometry Prove that meidians of triangle concurrent
Q06 |
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Q07 |
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Q08 |
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Q09 |
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Q10 |
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Answers
Q01. What is centroid
Definition
Medians of triangle are concurrent at a point G which is called centroid
Centroid is also called gravity center
What is concurrent ?
Three lines meet at one point is called concurrent
What is median ?
Vertex of triangle to mid point of opposite side is called median
diagram
Diagram of geometry
Program 03 02
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Q02. Medians of triangle are concurrent
diagram
Diagram of geometry
Program 21 09 : Diagram 09 02
Construction
Draw triangle ABC
Let D be mid point of BC, E be midpoint of CA and F be mid point of AB
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
Proof
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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Q03. 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
Conditions
Let A and B are fixed and C is moving with angle ACB = constant
Find locus of centroid
Diagram
Diagram of geometry
Program 10 03
Proof
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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Q05. Coordinate geometry Prove that medians of triangle concurrent
Reference
Coordinate geometry
Program 11 05
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Q06. Answer
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Q07. Answer
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Q08. Answer
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Q09. Answer
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Q10. Answer
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