Centroid of a Triangle

Q.

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

Q.

The centroid of a triangle lies at the origin and the coordinates of its two vertices are $\left(-8,7\right)$ and $\left(9,4\right)$. The area of the triangle is

1. $\frac{95}{6}$

2. $\frac{-285}{2}$

3. $\frac{190}{3}$

4. $285$

Q. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=k$, meets the coordinate axes at A, B, C such that the centroid of the triangle ABC is the point (a, b, c). Then k is

1. 3
2. 2
3. 1
4. 5

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.

A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A, B, C. The locus of the centroid of triangle ABC is

1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$

2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{9p^2}$

3. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{p}$

4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$

Q. If $z_1,z_2,z_3$ represent the vertices of an equilateral triangle such that $|z_1|=|z_2|=|z_3|$, then

1. 
2. 
3. 
4. 

Q.

If the coordinate of points $P(2,3,4),Q(1,-2,1)$, and $O$ is the origin then:

1. $OP=OQ$

2. $OP\perp OQ$

3. $OP\parallel OQ$

4. None of these

Q. If the position vector of the centroid of tetrahedron whose position vector of vertices are $\hat{i}+\hat{j}+3\hat{k},2\hat{i}-\hat{j},-3\hat{i}+2\hat{j}+8\hat{k}$ and $3\hat{i}+2\hat{j}+\hat{k}$ is $x\hat{i}+y\hat{j}+z\hat{k}.$ Then the value of $4(x-y+z)=$

Q. The co-ordinates of the foot of the perpendicular drawn from the point A(1, 0, 3)to the join of the points B(4, 7, 1) and C(3, 5, 3) are :

1. $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$
2. $(5,7,17)$
3. $(\frac{5}{7},-\frac{7}{3},\frac{17}{3})$
4. $(-\frac{5}{7},\frac{7}{3},-\frac{17}{3})$

Q.

If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3),

B(-2, b -5) and C(4, 7, C), find the values of a, b, c.

Q.

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.

Q.

The plane $ax+by+cz=1$ meets the coordinates axes at A, B and C. The centroid of $∆ABC$ is

1. $(3a,3b,3c)$

2. $\left(\frac{a}{3},\frac{b}{3},\frac{c}{3}\right)$

3. $\left(\frac{1}{3a},\frac{1}{3b},\frac{1}{3c}\right)$

4. $(a,b,c)$

Q. A plane meets the coordinate axes at points $A,B,C$ such that the centroid of the $△ABC$ is $(\alpha ,\beta ,\gamma )$. Show that the equation of the plane is $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$ .

Q.

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)

Q. In a three dimensional co - ordinate system P, Q and R are images of a point A(a, b, c) in the xy the yz and the zx planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is (O is the origin)

1. $0$
2. $a^2+b^2+c^2$
3. $\frac{2}{3}(a^2+b^2+c^2)$
4. $\frac{1}{2}(a^2+b^2+c^2)$

Q.

Consider an 'n' sided regular polygon with the origin as its centre. If $z_1$ be the complex number representing a vertex $a_1$ of the polygon, then the number associated with the vertex that is adjacent to $a_1$ is

1. 

2. 

3. 

4. 

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.

A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A, B, C. The locus of the centroid of triangle ABC is

1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$

2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{9p^2}$

3. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{p}$

4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$

Q. If Origin is the centroid of a triangle A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), then values of a, b and c will be respectively.

1. -2, -8, 2
2. 2, 8, 2
3. -2, 8, 2
4. 2, 8, -2

Q. A plane moves such that its distance from the origin is a constant p. If it intersects the coordinate axes at A, B, C then the locus of the centroid of the triangle ABC is

1. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$
2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{9}{p^2}$
3. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{2}{p^2}$
4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{4}{p^2}$

Q.

If a plane meets the coordinate axes at $\text{A, B}$ and $C$ such that the centroid of the triangle is $\text{(1, 2, 4),}$ then the equation of the plane is

1. $\text{x + 2y + 4z = 12}$

2. $\text{4x + 2y + z = 12}$

3. $\text{x + 2y + 4z = 3}$

4. $\text{4x + 2y + z = 3}$

Q. $D(2,1,0),E(2,0,0),F(0,1,0)$ are mid point of the sides $BC,CA,AB$ of $\Delta$ $ABC$ respectively, The the centroid of $\Delta$ABC is

1. $\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$
2. $\left(\frac{4}{3},\frac{2}{3},0\right)$
3. $(-\frac{1}{3},\frac{1}{3},\frac{1}{3})$
4. $\left(\frac{2}{3},\frac{1}{3},\frac{1}{3}\right)$

Q. The four lines drawing from the vertices of any tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is ‘k’ times the distance from each vertex to the opposite face, where k is

1. $\frac{1}{3}$
2. $\frac{3}{4}$
3. $\frac{1}{2}$
4. $\frac{5}{4}$

Q.

The coordinates of the mid-points of sides AB, BC and CA of $△$ are D (1, 2, -3) E(3, 0, 1) and F(-1, 1, -4) respectively. Write the coordinates of its centroid.

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. A plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(p,q,r)$. The equation of the plane is

1. none of these
2. $\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=0$
3. $\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=2$
4. $\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1$

Q. A tetrahedron is a three dimensional figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points A, B, C, D as its vertices which have coordinates $(x1,y1,z_1)(x_2,y_2,z_2)$ , $(x_3,y_3,z_3)$ and $(x_4,y_4,z_4)$, respectively in a rectangular three dimensional space. Then, the coordinates of its centroid are $[\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4},\frac{z_1+z_2+z_3+z_4}{4}]$.
Let a tetrahedron have three of its vertices represented by the points $(0,0,0),(6,5,1)$ and $(4,1,3)$ and its centroid lies at the point $(1,2,5)$. Now, answer the following question. The coordinate of the fourth vertex of the tetrahedron is:

1. $(-6,2,16)$
2. $(1,-2,13)$
3. $(-2,4,-2)$
4. $(1,-1,1)$

Q. Let vertices of a $△ABC$ are $A(-3,2,0),B(-2,0,2)$ and $C(1,-1,0).$ If $D$ is a point on $BC$ and $AD$ bisects the angle $\angle BAC,$ then point $D$ is

1. $(-\frac{7}{8},-\frac{3}{8},-\frac{10}{8})$
2. $(\frac{7}{8},\frac{3}{8},-\frac{10}{8})$
3. $(-\frac{7}{8},-\frac{3}{8},\frac{10}{8})$
4. $(\frac{7}{8},\frac{3}{8},\frac{10}{8})$

Q. If the centroid of the triangle with vertices $(3,a),(b,-8)$ and $(1,3)$ is $(3,-1)$ then find values of $a$ and $b$.

Q.

A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A, B, C. The locus of the centroid of triangle ABC is

1. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{3p}$

2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{9p^2}$

3. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{p}$

4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$