Explanation Given: The medians of a triangle meet at a point (the centroid) located two-thirds of the distance from each vertex to the midpoint of the opposite side. Proof: Assume PQRS is a tetrahedron. Now, consider the equilateral triangle PQR . Let T is the midpoint of the side QR . Hence the vector coordinates of point T = Q + R 2 . Now, G is the centroid of the triangle PQR which divides the vertex p and vertex T in the ratio 2:1. Hence, the vector coordinates of G = 2 T + P 2 + 1 = 2 T + P 3 Replace T = Q + R 2 in this equation to get; G = 2 T + P 3 = 2 ( Q + R 2 ) + P 3 = Q + R + P 3 Now, consider the equilateral triangle QRS Let H is the centroid of the triangle QRS . Therefore, the vector coordinates of the point H = S + Q + R 3 Now, assume that the point X is the centroid of the tetrahedron. And this point X divides the line which joins any vertex of the tetrahedron to its opposite face in the ratio m : 1 . Therefore, point X divides the line segment joining point S and the centroid of its opposite face PQR , that is, G in the ratio m : 1 . Hence, the vector coordinates of point X = m G + S m + 1 As point X also divides the line segment joining point P and the centroid of its opposite face QRS ,that is, H in the ratio m : 1 . Hence, the vector coordinates of point X = m H + P m + 1 Since both the equations represent the coordinates of the same point X . Hence, m G + S m + 1 = m H + P m + 1 Substitute G = Q + R + P 3 and H = S + Q + R 3 in the above equation to get m ( Q + R + P 3 ) + S m + 1 = m ( S + Q + R 3 ) + P m + 1 Multiply by ( m + 1 )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9, Problem 86SP

To determine

To prove:the four lines which join each vertex of the tetrahedron to the centroid of the opposite face meet at a point located three-fourths of the distance from the vertex to the centroid.

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Chapter 9, Problem 85SP

Chapter 9, Problem 87SP

Chapter 9 Solutions

Calculus

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To prove: the four lines which join each vertex of the tetrahedron to the centroid of the opposite face meet at a point located three-fourths of the distance from the vertex to the centroid.

Solution Summary: The author explains how the four lines which join each vertex of the tetrahedron to the centroid of its opposite face meet at a point located three-fourths of their distance.

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To prove:the four lines which join each vertex of the tetrahedron to the centroid of the opposite face meet at a point located three-fourths of the distance from the vertex to the centroid.

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Chapter 9 Solutions