Question:

ABC is a triangle and AD is the median. If the coordinates of A are ( 4, 7, - 8)and the coordinates of centroid of the triangle ABC are (1, 1, 1), what are the coordinates of D?

Updated On: Jul 6, 2022

$\left(- \frac{1}{2}, 2 ,11\right)$
$\left(- \frac{1}{2}, -2 ,\frac{11}{2} \right)$
(-1, 2, 11)
(-5, -11, 19)

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The Correct Option is B

Solution and Explanation

Let coordinates of D be (x, y, z) Co-ordinates of centroid is (1, 1, 1), and of A, is (4, 7, 8) Centroid divides median is 2 : 1 ratio So, $\frac{AO}{OD} = 2 : 1$ For x : $1 = \frac{2 \times x + 1 \times 4}{1 + 2}$ $\Rightarrow \, x = - 12$ For y : $1 = \frac{2y + 1 \times 7}{ 1+ 2}$ $y = - 2$ $1 = \frac{2 \times z + 1 \times -8 }{3} \, \Rightarrow \, z = + 11/2$ $\therefore$ Co-ordinates of D are (-1/2, -2, 11/2)

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Notes on introduction to three dimensional geometry

Concepts Used:

Introduction to Three Dimensional Geometry

In mathematics, Geometry is one of the most important topics. The concepts of Geometry are defined with respect to the planes. So, Geometry is divided into three categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.

Direction Cosines and Direction Ratios of Line

Let's consider line ‘L’ is passing through the three-dimensional plane. Now, x,y, and z are the axes of the plane, and α,β, and γ are the three angles the line making with these axes. These are called the plane's direction angles. So, correspondingly, we can very well say that cosα, cosβ, and cosγ are the direction cosines of the given line L.