Advertisements
Advertisements
Question
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
Solution
Let the coordintes of the triangles be A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3).
Now,
Mid point of AB is D(1, 2, −3)
\[\frac{x_1 + x_2}{2} = 1, \frac{y_1 + y_2}{2} = 2 \text{ and } \frac{z_1 + z_2}{2} = - 3\]
\[ \Rightarrow x_1 + x_2 = 2, y_1 + y_2 = 4 \text{ and } z_1 + z_2 = - 6 . . . . . \left( 1 \right)\]
Mid point of BC is E(3, 0,1)
\[\frac{x_2 + x_3}{2} = 3, \frac{y_2 + y_3}{2} = 0 \text{ and } \frac{z_2 + z_3}{2} = 1\]
\[ \Rightarrow x_2 + x_3 = 6, y_2 + y_3 = 0 \text{ and } z_2 + z_3 = 2 . . . . . \left( 2 \right)\]
Mid point of AC is F(−1, 1, −4)
\[\frac{x_1 + x_3}{2} = - 1, \frac{y_1 + y_3}{2} = 1 \text{ and } \frac{z_1 + z_3}{2} = - 4\]
\[ \Rightarrow x_1 + x_3 = - 2, y_1 + y_3 = 2 \text{ and } z_1 + z_3 = - 8 . . . . . \left( 3 \right)\]
Adding (1), (2) and (3),we get
\[2\left( x_1 + x_2 + x_3 \right) = 6, 2\left( y_1 + y_2 + y_3 \right) = 6 \text{ and } 2\left( z_1 + z_2 + z_3 \right) = - 12\]
\[ \Rightarrow x_1 + x_2 + x_3 = 3, y_1 + y_2 + y_3 = 3 \text{ and } z_1 + z_2 + z_3 = - 6\]
\[ \Rightarrow \frac{x_1 + x_2 + x_3}{3} = 1, \frac{y_1 + y_2 + y_3}{3} = 1 \text{ and } \frac{z_1 + z_2 + z_3}{3} = - 2\]
Hence, the centroid of the traingle ABC is (1, 1, −2).
APPEARS IN
RELATED QUESTIONS
Coordinate planes divide the space into ______ octants.
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Write the distance of the point P(3, 4, 5) from z-axis.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
What is the locus of a point for which y = 0, z = 0?
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
The perpendicular distance of the point P(3, 3,4) from the x-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/γ` = 3
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
`1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
Find the foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
Show that the points `(hati - hatj + 3hatk)` and `3(hati + hatj + hatk)` are equidistant from the plane `vecr * (5hati + 2hatj - 7hatk) + 9` = 0 and lies on opposite side of it.
If the directions cosines of a line are k, k, k, then ______.
The locus represented by xy + yz = 0 is ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
