Centroid of a Triangle

Q.

Which of the following statements are correct?

1. For triangle orthocenter, circumcenter and

centroid are collinear.

2. Centroid divides the line joining the circumcenter $\&$ orthocenter

in the ratio of $2:1$ i.e., $\frac{CS}{OC}=\frac{2}{1}$

Where C-Coordinate of centroid

O-Coordinate of orthocenter

S--Coordinate of circumcenter

1. Both 1 & 2

2. None of these

3. Only 1

4. Only 2

Q.

If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

1. (-3, 3)

2. (3, 3)

3. (3, 1)

4. (1, 3)

Q.

Find the coordinates of the points which trisect the line segment AB, if two points are A(2, 1, -3) and B (5, -8, 3).

Q.

Show that the coordinates of the centroid of the triangle with vertices $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(x_3,y_3,z_3)$ is $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3})$.

Q.

Find the ratio in which the line joining A(2, 1, 5) and B(3, 4, 3) is divided by the plane 2x+2y-2z=1. Also, find the coordinates of the point of division.

Q.

Find the centroid of a triangle, mid-points of whose sides are (1, 2, -3), (3, 0, 1) and (- 1, 1, - 4).

Or

The mid-points of the sides of a triangle are (1, 5, - 1), (0, 4, - 2) and (2, 3, 4). Find its vertices.

Q.

In a triangle coordinates of orthocenter and circumcenter are $(-3,5,2)$ and $(6,2,5)$. Find the coordinates of centroid of the triangle.

1. $(2,2,2)$

2. $(3,3,3)$

3. $(2,2,4)$

4. $(3,3,4)$

Q. If the origin is the centroid of the triangle PQR with vertices P (2 a , 2, 6), Q (–4, 3 b , –10) and R (8, 14, 2 c ), then find the values of a , b and c .

Q. if A(cos alpha, sin alpha) , B(sin alpha, - cos alpha) and C (2, 1) are vertices of a triangle. then the locus of its centroid if alpha varies is:

Q.

The centroid of a triangle, whose vertices are (2, 1), (5, 2) and (3, 4), is

1. 

2. 

3. 

4. 

Q.

If the middle points of the sides of a triangle be (-2, 3), (4, -3) and (4, 5), then the centroid of the triangle is

1. (5/3, 2)

2. (5/6, 1)

3. (2, 5/3)

4. (1, 5/6)

Q. 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.

Q. Let $A(3,2,0),B(5,3,2),C(-9,6,-3)$ be three points forming a triangle. $AD$, the bisectors of $\angle BAC$, meets $BC$ an $D$. Find the coordinates of the point $D$.

Q.

If the origin is the centroid of the triangle ABC with vertices A (2a, 14, 6), B (8, 3b, -10) and C(-4, 2, 2c), then the values of a and c are.

1. 2 and -2

2. 

3. -2 and 2

4. 

Q.

If the origin is the centriod 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.

Q.

In an equilateral $△ABC$ coordinates of the centroid of the triangle is ($1,2,3)$. Find the coordinates of the orthocenter of the triangle.

1. $(1,2,3)$

2. $(\frac{1}{2},1,\frac{3}{2})$

3. $(1,2,\infty )$

4. $(0,2,3)$

Q.

If the centroid of triangle whose vertices are (a, 1, 3), (– 2, b, –5) and (4, 7, c) be the origin, then the values of a, b, c are

1. – 2, –8, –2

2. 2, 8, –2

3. –2, –8, 2

4. 7, –1, 0

Q.

Which of the following statements are correct?

1. For triangle orthocenter, circumcenter and

centroid are collinear.

2. Centroid divides the line joining the circumcenter $\&$ orthocenter

in the ratio of $2:1$ i.e., $\frac{CS}{OC}=\frac{2}{1}$

Where C-Coordinate of centroid

O-Coordinate of orthocenter

S--Coordinate of circumcenter

1. Only 1

2. Only 2

3. Both 1 & 2

4. None of these

Q.

A plane meets the coordinate axes at points A, B, C and $(\alpha ,\beta ,\gamma )$ is the centroid of the triangle ABC. Then the equation of the plane is

1. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$

2. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$

3. $\frac{3x}{\alpha }+\frac{3y}{\beta }+\frac{3z}{\gamma }=1$

4. $\alpha x+\beta y+\gamma z=1$

Q. If the centroid of triangle whose vertices are $(a,1,3),(-2,b,-5)$ and $(4,7,c)$ be the origin, the value of $c-a-b$ is___