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Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4?
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Ab square+BC square+AC square=k(AG square+BG square+CG square), where ...
Solution:
Given:
AB square + BC square + AC square = k(AG square + BG square + CG square)
Centroid of a Triangle:
The centroid of a triangle is the point where the three medians intersect. In a triangle ABC, the centroid G divides each median in the ratio 2:1.
Properties of Centroid:
1. The centroid divides each median into segments of 2:1 ratio.
2. The centroid is located two-thirds of the distance from each vertex along the median.
Proof:
Let D, E, F be the midpoints of sides BC, AC, AB respectively.
From the median theorem,
AD = 2/3 * AG,
BD = 2/3 * BG,
CD = 2/3 * CG.
Now, we have
AG = 3/2 * AD,
BG = 3/2 * BD,
CG = 3/2 * CD.
Substitute these values in the given equation,
AB square + BC square + AC square = k[(3/2 * AD) square + (3/2 * BD) square + (3/2 * CD) square]
Now simplify the equation to find the value of k.
Conclusion:
After simplifying the equation, we get k = 3. Therefore, the correct option is 3) 3.
Given:
AB square + BC square + AC square = k(AG square + BG square + CG square)
Centroid of a Triangle:
The centroid of a triangle is the point where the three medians intersect. In a triangle ABC, the centroid G divides each median in the ratio 2:1.
Properties of Centroid:
1. The centroid divides each median into segments of 2:1 ratio.
2. The centroid is located two-thirds of the distance from each vertex along the median.
Proof:
Let D, E, F be the midpoints of sides BC, AC, AB respectively.
From the median theorem,
AD = 2/3 * AG,
BD = 2/3 * BG,
CD = 2/3 * CG.
Now, we have
AG = 3/2 * AD,
BG = 3/2 * BD,
CG = 3/2 * CD.
Substitute these values in the given equation,
AB square + BC square + AC square = k[(3/2 * AD) square + (3/2 * BD) square + (3/2 * CD) square]
Now simplify the equation to find the value of k.
Conclusion:
After simplifying the equation, we get k = 3. Therefore, the correct option is 3) 3.
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Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4?
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Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4?.
Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Ab square+BC square+AC square=k(AG square+BG square+CG square), where G is centroid of the triangle ABC,then k=. Options are. 1)1. 2)2. 3)3. 4)4?.
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