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# Solution - By vector method prove that the medians of a triangle are concurrent. - HSC Science (General) 12th Board Exam - Mathematics and Statistics

ConceptMedians of a Triangle Are Concurrent

#### Questions

By vector method prove that the medians of a triangle are concurrent.

Using vector method prove that the medians of a triangle are concurrent.

#### Solution

Let A, B and C be the vertices of a triangle.
Let D, E and F be the midpoints of the sides BC, AC and AB respectively.

Let bar(GA) = bara,bar(GB) = barb,bar(GC) = barc,bar(GD) = bard,bar(GE) =bar e and bar(GF) =bar f  be position vectors
of points A, B, C, D, E and F respectively.
Therefore, by midpoint formula,

bard=(barb+barc)/2,bare=(bara+barc)/2 and barf=(bara+barb)/2

2bard=barb+barc,2bare=bara+barc and 2bar f=bara+barb

2bard+bara=bara+barb+barc ,2bare+barb=bara+barb+barc and 2barf+barc=bara+barb+barc

(2bard+bara)/3=(2bare+barb)/3=(2barf+barc)/3=(bara+barb+barc)/3

Let " " barg=(bara+barb+barc)/3

therefore " We have " barg=(bara+barb+barc)/3=((2)bard+(1)bara)/3=((2)bare+(1)barb)/3=((2)barf+(1)barc)/3

If G is the point whose position vector is  barg , then from the above equation it is clear that the point G lies on the medians AD, BE, CF
and it divides each of the medians AD, BE, CF internally in the ratio 2:1. Therefore, three medians are concurrent.

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#### APPEARS IN

2015-2016 (March) (with solutions)
Question 2.1.2 | 3 marks
2017-2018 (March) (with solutions)
Question 2.1.1 | 3 marks

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Solution for question: By vector method prove that the medians of a triangle are concurrent. concept: null - Medians of a Triangle Are Concurrent. For the courses HSC Science (General) , HSC Arts, HSC Science (Electronics), HSC Science (Computer Science)
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