Advertisements
Advertisements
प्रश्न
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
Advertisements
Notes
\[\text{ Let A }\left( x_1 , y_1 , z_1 \right), B\left( x_2 , y_2 , z_2 \right) \text{ and } C\left( x_3 , y_3 , z_3 \right) \text{ be the vertices of the given triangle }, \]
\[\text{ and let } D\left( 1, 2, - 3 \right) , E\left( 3, 0, 1 \right) \text{ and } F\left( - 1, 1, - 4 \right) \text{ be the mid points of the sides BC, CA and AB } respectively . \]
\[\text{ D is the mid point of BC }\]
\[ \therefore \frac{x_2 + x_3}{2} = 1, \frac{y_2 + y_3}{2} = 2 \text{ and } \frac{z_2 + z_3}{2} = - 3\]
\[ \Rightarrow x_2 + x_3 = 2, y_2 + y_3 = 4 \text{ and } z_2 + z_3 = - 6 . . . \left( 1 \right)\]
\[\text{ E is the mid point of CA }\]
\[ \therefore \frac{x_1 + x_3}{2} = 3, \frac{y_1 + y_3}{2} = 0 \text{ and } \frac{z_1 + z_3}{2} = 1\]
\[ \Rightarrow x_1 + x_3 = 6, y_1 + y_3 = 0 \text{ and } z_1 + z_3 = 2 . . . \left( 2 \right)\]
\[\text{ F is the mid point of AB }\]
\[ \therefore \frac{x_1 + x_2}{2} = - 1, \frac{y_1 + y_2}{2} = 1 \text{ and } \frac{z_1 + z_2}{2} = - 4\]
\[ \Rightarrow x_1 + x_2 = - 2, y_1 + y_2 = 2 \text{ and } z_1 + z_2 = - 8 . . . \left( 2 \right)\]
\[\text{ Adding first three equations we get }, \]
\[2\left( x_1 + x_2 + x_3 \right) = 6, 2\left( y_1 + y_2 + y_3 \right) = 6 \text{ and } 2\left( z_1 + z_2 + z_3 \right) = - 12\]
\[ \Rightarrow x_1 + x_2 + x_3 = 3, y_1 + y_2 + y_3 = 3 \text{ and } z_1 + z_2 + z_3 = - 6\]
\[\text{ The coordinate of the centroid is given by } \]
\[\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right)\]
\[ \Rightarrow \left( 1, 1, - 2 \right)\]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the distance between the pairs of points:
(2, 3, 5) and (4, 3, 1)
Find the distance between the following pairs of points:
(–1, 3, –4) and (1, –3, 4)
Find the distance between the following pairs of points:
(2, –1, 3) and (–2, 1, 3)
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are the vertices of an isosceles triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the distance between the following pairs of point:
A(3, 2, –1) and B(–1, –1, –1).
Using distance formula prove that the following points are collinear:
P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)
Using distance formula prove that the following points are collinear:
A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)
Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus.
Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.
If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a.
Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1).
Find the distance of the point whose position vector is `(2hati + hatj - hatk)` from the plane `vecr * (hati - 2hatj + 4hatk)` = 9
Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vecr = 2hati - hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr * (hati - hatj + hatk)` = 5.
The distance of a point P(a, b, c) from x-axis is ______.
Find the angle between the lines `vecr = 3hati - 2hatj + 6hatk + lambda(2hati + hatj + 2hatk)` and `vecr = (2hatj - 5hatk) + mu(6hati + 3hatj + 2hatk)`
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
Find the shortest distance between the lines given by `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk` and `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
Find the equation of the plane through the intersection of the planes `vecr * (hati + 3hatj) - 6` = 0 and `vecr * (3hati + hatj + 4hatk)` = 0, whose perpendicular distance from origin is unity.
Distance of the point (α, β, γ) from y-axis is ______.
The distance of the plane `vecr * (2/4 hati + 3/7 hatj - 6/7hatk)` = 1 from the origin is ______.
If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord of another circle 'C' whose center is at (2, 1), then its radius is ______.
