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.
Advertisements
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.
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(5, 2, –7) in the xy-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.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
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.
Write the distance of the point P (2, 3,5) from the xy-plane.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
What is the locus of a point for which y = 0, z = 0?
The length of the perpendicular drawn from the point P(a, b, c) from z-axis is
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
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.
Find the image of the point having position vector `hati + 3hatj + 4hatk` in the plane `hatr * (2hati - hatj + hatk)` + 3 = 0.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
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
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
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.
If l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
If the directions cosines of a line are k, k, k, then ______.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The unit vector normal to the plane x + 2y +3z – 6 = 0 is `1/sqrt(14)hati + 2/sqrt(14)hatj + 3/sqrt(14)hatk`.
