Points of Concurrency in a Triangle

Medians of a Triangle are Concurrent. The Point of Concurrency is the Centroid.

A median of a triangle is the line joining the midpoint of a side and the opposite vertex.

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Angle Bisectors of a Triangle are Concurrent. The Point of Concurrency is the Incenter.

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Altitudes of a Triangle are Concurrent. The Point of Concurrency is the Orthocenter.

An altitude is the perpendicular line drawn from a vertex on the opposite side of the triangle.

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Perpendicular Bisectors of a Triangle are Concurrent. The Point of Concurrency is the Circumcenter.

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Volume of Parallelopiped and Tetrahedron

A parallelopiped is a 3-D object, each of whose faces is a parallelogram.

A rectangular parallelopiped is the one, whose faces are rectangular.

A cube is a rectangular parallelopiped, whose edges are of equal length.

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A tetrahedron is a 3-D object, whose all faces are triangular. It can be shown that a parallelopiped can be decomposed into 6 tetrahedra.

Image Source : https://www.dune-project.org/

For a visual proof :  1 Parallelopipded  = 6 Tetrahedra


 

Let there be 2 such objects, whose coterminus edges are identical. Then, the volume of the tetrahedron is {\dfrac 1 6}th of that of the parallelopiped. The expression can also be obtained using the following theorem :

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Vector Algebra : Theorems (I)

In this blog, we will prove fundamental theorems in vector algebra. These theorems are in accordance with the +2 curriculum.

Theorem 1 : Collinear Vectors

vectors_theorems01-00.jpgTheorem 2 : Non-collinear and Coplanar Vectors

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Theorem 3 : Coplanar Vectors

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Theorem 4 : Non-coplanar Vectors

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